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SUMMARY TECHNICAL REPORT 
OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 


Manuscript and illustrations for this volume were prepared for 
publication by the Summary Reports Groups of the Columbia 
University Division of War Research under contract OEMsr-1131 
with the Office of Scientific Research and Development. This vol¬ 
ume was printed and bound by the Columbia University Press. 

Distribution of the Summary Technical Report of NDRC has been 
made by the War and Navy Departments. Inquiries concerning the 
availability and distribution of the Summary Technical Report 
volumes and microfilmed and other reference material should be 
addressed to the War Department Library, Room 1A-522, The 
Pentagon, Washington 25, D. C., or to the Office of Naval Re¬ 
search, Navy Department, Attention: Reports and Documents 
Section, Washington 25, D. C. 


Copy No. 

6 


This volume, like the seventy others of the Summary Technical 
Report of NDRC, has been written, edited, and printed under 
great pressure. Inevitably there are errors which have slipped past 
Division readers and proofreaders. There may be errors of fact not 
known at time of printing. The author has not been able to follow 
through his writing to the final page proof. 

Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 

WASHINGTON 25, D. C. 

A master errata sheet will be compiled from these reports and sent 
to recipients of the volume. Your help will make this book more 
useful to other readers and will be of great value in preparing any 
revisions. 


SUMMARY TECHNICAL REPORT OF DIVISION 6, NDRC 


VOLUME 8 


THE PHYSICS OF SOUND IN 

THE SEA 


OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

DIVISION 6 
JOHN T. TATE, CHIEF 


WASHINGTON, D. C., 1946 



NATIONAL DEFENSE RESEARCH COMMITTEE 


James B. Conant, Chairman 
Richard C. Tolman. Vice Chairman 


Roger Adams 
Frank B. Jewett 
Karl T. Compton 

Irvin Stewart, 


Army Representative 1 
Navy Representative 2 
Commissioner of Patents 3 
Executive Secretary 


1 Army representatives in order of service: 


Maj. Gen. 
Maj. Gen. 
Maj. Gen. 
Brig. Gen. 


G. V. Strong 
R. C. Moore 
C. C. Williams 
W. A. Wood, Jr. 


Col. L. A. Denson 
Col. P. R. Faymonville 
Brig. Gen. E. A. Regnier 
Col. M. M. Irvine 


Col. E. A. Routhea 


-Navy representatives in order of service: 

Rear Adm. H. G. Bowen Rear Adm. J. A. Purer 

Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 

Commodore H. A. Schade 
3 Commissioners of Patents in order of service: 

Conway P. Coe Casper W. Ooms 


NOTES ON THE ORGANIZATION OF NDRC 


The duties of the National Defense Research Committee 
were (1) to recommend to the Director of OSRD suitable 
projects and research programs on the instrumentalities of 
warfare, together with contract facilities for carrying out 
these projects and programs, and (2) to administer the tech¬ 
nical and scientific work of the contracts. More specifically, 
NDRC functioned by initiating research projects on re¬ 
quests from the Army or the Navy, or on requests from an 
allied government transmitted through the Liaison Office 
of OSRD, or on its own considered initiative as a result of 
the experience of its members. Proposals prepared by the 
Division, Panel, or Committee for research contracts for 
performance of the work involved in such projects were 
first reviewed by NDRC, and if approved, recommended to 
the Director of OSRD. Upon approval of a proposal by the 
Director, a contract permitting maximum flexibility of 
scientific effort was arranged. The business aspects of the 
contract, including such matters as materials, clearances, 
vouchers, patents, priorities, legal matters, and administra¬ 
tion of patent matters were handled by the Executive Sec¬ 
retary of OSRD. 

Originally NDRC administered its work through five 
divisions, each headed by one of the NDRC members. 
These were: 

Division A — Armor and Ordnance 

Division B — Bombs, Fuels, Gases, & Chemical Problems 
Division C — Communication and Transportation 
Division D — Detection, Controls, and Instruments 
Division E — Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three ad¬ 
ministrative divisions, panels, or committees were created, 
each with a chief selected on the basis of his outstanding 
work in the particular field. The NDRC members then be¬ 
came a reviewing and advisory group to the Director of 
OSRD. The final organization was as follows: 

Division 1 — Ballistic Research 

Division 2 — Effects of Impact and Explosion 

Division 3 — Rocket Ordnance 

Division 4 — Ordnance Accessories 

Division 5 — New Missiles 

Division 6 — Sub-Surface Warfare 

Division 7 — Fire Control 

Division 8 — Explosives 

Division 9 — Chemistry 

Division 10 — Absorbents and Aerosols 

Division 11 — Chemical Engineering 

Division 12 — Transportation 

Division 13 — Electrical Communication 

Division 14 — Radar 

Division 15 — Radio Coordination 

Division 16 — Optics and Camouflage 

Division 17 — Physics 

Division 18 — War Metallurgy 

Division 19 — Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deterioration Administrative Committee 


Library of Congress 



iv 

66 


2015 


490953 





























NDRC FOREWORD 


A s events of the years preceding 1940 revealed 
more and more clearly the seriousness of the 
world situation, many scientists in this country came 
to realize the need of organizing scientific research for 
service in a national emergency. Recommendations 
which they made to the White House were given care¬ 
ful and sympathetic attention, and as a result the 
National Defense Research Committee [NDRC] was 
formed by Executive Order of the President in the 
summer of 1940. The members of NDRC, appointed 
by the President, were instructed to supplement the 
work of the Army and the Navy in the development 
of the instrumentalities of war. A year later, upon 
the establishment of the Office of Scientific Research 
and Development [OSRD], NDRC became one of 
its units. 

The Summary Technical Report of NDRC is a 
conscientious effort on the part of NDRC to sum¬ 
marize and evaluate its work and to present it in a 
useful and permanent form. It comprises some 
seventy volumes broken into groups corresponding 
to the NDRC Divisions, Panels, and Committees. 

The Summary Technical Report of each Division, 
Panel, or Committee is an integral survey of the work 
of that group. The first volume of each group’s re¬ 
port contains a summary of the report, stating the 
problems presented and the philosophy of attacking 
them and summarizing the results of the research, de¬ 
velopment, and training activities undertaken. Some 
volumes may be “state of the art” treatises covering 
subjects to which various research groups have con¬ 
tributed information. Others may contain descrip¬ 
tions of devices developed in the laboratories. A 
master index of all these divisional, panel, and com¬ 
mittee reports which together constitute the Sum¬ 
mary Technical Report of NDRC is contained in a 
separate volume, which also includes the index of a 
microfilm record of pertinent technical laboratory 
reports and reference material. 

Some of the NDRC-sponsored researches which 
had been declassified by the end of 1945 were of 
sufficient popular interest that it was found desirable 
to report them in the form of monographs, such as 
the series on radar by Division 14 and the monograph 
on sampling inspection by the Applied Mathematics 
Panel. Since the material treated in them is not dupli¬ 


cated in the Summary Technical Report of NDRC, 
the monographs are an important part of the story 
of these aspects of NDRC research. 

In contrast to the information on radar, which is 
of widespread interest and much of which is released 
to the public, the research on subsurface warfare is 
largely classified and is of general interest to a more 
restricted group. As a consequence, the report of 
Division 6 is found almost entirely in its Summary 
Technical Report, which runs to over twenty volumes. 
The extent of the work of a Division cannot therefore 
be judged solely by the number of volumes devoted 
to it in the Summary Technical Report of NDRC: 
account must be taken of the monographs and avail¬ 
able reports published elsewhere. 

Any great cooperative endeavor must stand or fall 
with the will and integrity of the men engaged in it. 
This fact held true for NDRC from its inception, and 
for Division 6 under the leadership of Dr. John T. 
Tate. To Dr. Tate and the men who worked with 
him — some as members of Division 6, some as 
representatives of the Division’s contractors — be¬ 
longs the sincere gratitude of the Nation for a diffi¬ 
cult and often dangerous job well done. Their efforts 
contributed significantly to the outcome of our naval 
operations during the war and richly deserved the 
warm response they received from the Navy. In ad¬ 
dition, their contributions to the knowledge of the 
ocean and to the art of oceanographic research will 
assuredly speed peacetime investigations in this field 
and bring rich benefits to all mankind. 

The Summary Technical Report of Division 6, 
prepared under the direction of the Division Chief 
and authorized by him for publication, not only 
presents the methods and results of widely varied re¬ 
search and development programs but is essentially a 
record of the unstinted loyal cooperation of able men 
linked in a common effort to contribute to the defense 
of their Nation. To them all we extend our deep 
appreciation. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. Conant, Chairman 
National Defense Research Committee 











































































FOREWORD 


T his volume, together with Volumes 6, 7, and 9, 
summarizes four years of research on underwater 
sound phenomena. The purpose of this research was 
to provide a firmer foundation for the most effective 
design and use of sonar gear. It is generally true that 
wide basic knowledge is an important element in en¬ 
gineering practice. In the development of sonar gear, 
knowledge of how sound is generated, transmitted, 
reflected, received, and detected is clearly useful 
both in the design of new equipment and in the most 
efficient utilization of existing gear. As a result of the 
time delay between the design of new equipment and 
its use in service, the most important application of 
this basic information during World War II has been 
in suggesting how existing equipment could best be 
operated and tactically used. 

The importance of basic information on under¬ 
water sound had been evident to both our own Navy 
and the British for some time. Practical experience 
had shown that the maximum distance at which a 
target could be detected with underwater sound was 
highly variable, even when the equipment was in 
good operating condition. Since it was realized that 
such variability might well be related to a variability 
in oceanographic conditions, the Navy brought this 
problem to the attention of the Woods Hole Oceano¬ 
graphic Institution and the Scripps Institute of 
Oceanography. To support an investigation, NDRC 
contracted in 1940 with the former institution to 
carry out studies and experimental investigations of 
the structure of the superficial layer of the ocean and 
its effect on the transmission of sonic and supersonic 
vibrations. 

The work carried out under this contract, together 
with supporting information obtained elsewhere, em¬ 
phasized the relation of such basic factors to the 
variable performance of sonar gear. Thus when some 
months later it was proposed to establish a section in 
NDRC to undertake research and development re¬ 
lating to the detection of submerged submarines, 
plans were made to increase substantially this re¬ 
search effort. To this end, the plans which were for¬ 
mulated by NDRC and approved by the Navy in¬ 
cluded research on underwater sound phenomena at 
the proposed laboratory at San Diego, to be operated 
under a contract with the University of California 
Division of War Research. This step not only in¬ 


creased the number of personnel engaged in this re¬ 
search and facilitated study of oceanic conditions 
peculiar to the Pacific area, but also most fortunately 
made it possible for the San Diego Laboratory to 
recruit certain of its staff from the Scripps Institution 
of Oceanography and to draw upon the director and 
staff of the Scripps Institution for very pertinent 
background information in oceanography. While the 
major source of the experimental data continued to 
be the Woods Hole and San Diego Laboratories, very 
pertinent data were from time to time obtained from 
other laboratories, notably New London, Harvard, 
the Massachusetts Institute of Technology, and the 
Underwater Sound Reference Laboratories. 

Quite promptly, an analytical section, later known 
as the Sonar Analysis Group, was organized under a 
contract with Columbia University Division of War 
Research. The function of this group was to assist in 
the analyses of data being accumulated by Woods 
Hole, San Diego, and other laboratories, and, as it 
became possible to draw conclusions, to present these 
to other groups interested in operations or design. In 
this connection it should be emphasized that the 
seeming importance of this research to the Navy led 
to the assignment of naval personnel to follow the 
work actively. In particular, officers of the Sonar De¬ 
sign Section of the Bureau of Ships followed very 
closely the research of this analytical group, partici¬ 
pating directly in much of the work. 

The results obtained in this research and sum¬ 
marized in this and companion volumes found many 
important applications during World War II. The 
rules used for operating sonar gear were based in part 
on these results. Many tactical rules embodied in 
submarine and antisubmarine doctrine were directly 
based on information obtained in these basic studies 
of transmission, reflection, detection, and the like. 
As an example, the spacing between antisubmarine 
vessels in different tactical and oceanographic condi¬ 
tions was varied according to the measured tempera¬ 
ture gradients in the upper layers of the ocean. In 
addition, the choice of operating frequency, pulse 
length, size, and power for new equipment, especially 
for submarines, was considerably influenced by such 
basic knowledge. It can be stated with considerable 
confidence that a detailed basic knowledge of under¬ 
water sound phenomena will be of increasing help in 

vii 


FOREWORD 


the design and operation of Navy sonar equip¬ 
ment. 

Only a few of the scientists and others contribut- 
ingto this war effort can be named. Mr.C.O’D.Iselin, 
Director of the Woods Hole Oceanographic Institu¬ 
tion, and his staff brought to this research, to which 
they ably contributed, a sound background knowl¬ 
edge of oceanography. Dr. V. 0. Knudsen, Dean of 
the Graduate School of the University of California 
at Los Angeles and for some time the Director of the 
Division’s San Diego Laboratory, was one of this 
country’s foremost scientists in the field of acoustics. 
Dr. Knudsen played a prominent part in organizing 
the research program, and after leaving the San 
Diego Laboratory he contributed actively and ef¬ 
fectively to research work closely related to the sub¬ 
ject of this volume. Dr. G. P. Harnwell, Chairman of 
the Department of Physics at the University of 
Pennsylvania, who succeeded Dr. Knudsen as Di¬ 
rector at San Diego after having served some time as 
a technical aide to the Division, gave wise general 
direction to this research at San Diego. In operations 
at San Diego, Dr. Knudsen and Dr. Harnwell were 
ably supported by Dr. Carl Eekart, Professor of 
Theoretical Physics at the University of Chicago, 
who became Associate Director at San Diego, re¬ 
sponsible for the planning and execution of the basic 
research there. 

Dr. H. Sverdrup, Director of the Scripps Institu¬ 
tion of Oceanography, and his staff also contributed 
significantly to this work. The U. S. Navy Electronics 
Laboratory at San Diego collaborated most helpfully 
in much of this basic research. The task of organizing 
the very important analytical work was assumed by 
Dr. W. V. Houston, Professor of Physics at the Cali¬ 
fornia Institute of Technology and Director of the 
Special Studies Group; he delegated the very large 
part of the responsibility to Dr. Lyman Spitzer, Jr., 
an outstanding member of the Departments of 


Physics and Astronomy at Yale University, who be¬ 
came Director of the Sonar Analysis Group. 

As the reader will note, Dr. Spitzer undertook the 
responsibility for preparing this volume, and in this 
he had had the assistance not only of members of his 
own staff but also of naval personnel and of members 
of the Woods Hole and San Diego staffs. The Divi¬ 
sion appreciates the efforts of all those who have 
participated. 

This research project secured most effective sup¬ 
port from the Navy. The broad program of research 
and study which was proposed by Dr. Jewett and 
Dr. Bush, and which included this basic research on 
underwater sound, was supported by Rear Admiral 
S. M. Robinson, Chief of the Bureau of Ships, who 
took steps to provide facilities for this work at San 
Diego. Later, when Rear Admiral Van Keuren be¬ 
came Chief of the Bureau of Ships, he likewise 
strongly backed the program, which was still in its 
initial stages. Support of the program continued with 
Vice Admiral Cochrane as Chief of the Bureau of 
Ships, and most helpful liaison was provided by 
Captain Rawson Bennett, Jr., Commander J. C. 
Myers, Commander Roger Revelle, and others in the 
Bureau. The Coordinator of Research and Develop¬ 
ment and his staff continually gave support to this 
research. The results of much of this work were of 
special interest to the Tenth Fleet and very close 
contact was accordingly maintained with its staff, 
particularly with the Operations Research Group. 

In presenting this volume the hope is expressed 
that research in this area will be energetically con¬ 
tinued. It is also hoped that general interest in this 
field may be maintained by the distribution to the 
widest possible audience of this volume and other 
volumes which have been written from the stand¬ 
point of basic science. 

John T. Tate 
Chief, Division (i 



PREFACE 


I n the course of prosubmarine and antisubmarine 
research carried out during World War II, a large 
amount of information was obtained on the propaga¬ 
tion of underwater sound. Much of this was gathered 
in fairly random ways, such as while testing under¬ 
water sound equipment. Most of the useful informa¬ 
tion, however, was obtained by groups devoted pri¬ 
marily to the problem of underwater sound propaga¬ 
tion. While valuable results had been found before 
World War II by the Naval Research Laboratory, 
the British, and other groups, most of the informa¬ 
tion on underwater sound transmission obtained 
during the war resulted from a program of studies 
organized by Division 6 of the National Defense Re¬ 
search Committee and carried out in collaboration 
with Navy laboratories at San Diego and elsewhere. 

It should be kept in mind that these so-called 
fundamental programs were not fundamental in the 
usual scientific sense. They were not aimed at isolat¬ 
ing and understanding the different factors at work, 
but were designed rather for the accumulation of in¬ 
formation which would be useful in antisubmarine 
and prosubmarine operations. Thus effort was con¬ 
centrated on the study of the transmission loss of 
sound generated with standard sonar gear under 
varying oceanographic conditions, rather than on a 
detailed study of each of the individual factors af¬ 
fecting underwater sound transmission. Similarly, 
the reflection of sound from actual submarines was 
studied rather than the individual mechanisms re¬ 
sponsible for the origin of echoes from underwater 
targets. 

During the war this approach was abundantly 
justified by its results. The information obtained on 
underwater sound propagation under different ocea¬ 
nographic and tactical situations was immediately ap¬ 
plied to the more effective use of existing underwater 
sound equipment in different situations. The results 
of transmission, reverberation and other studies were 
usually used operationally much more rapidly than 
the results of equipment development. 

Over a longer period, however, information on 
underwater sound can be most useful il the phenom¬ 
ena are not merely observed but also explained. An 
understanding of each of the basic factors affecting 


underwater sound propagation would make it possi¬ 
ble to predict the transmission and reflection to be 
expected under conditions widely different from 
those prevailing when the original measurements 
were taken. While the primarily experimental re¬ 
search carried out during the war could be immedi¬ 
ately applied to the gear then in existence, the de¬ 
velopment of new equipment for new and unforeseen 
tactical situations requires an understanding of the 
factors which influence underwater sound. The ulti¬ 
mate aim of basic underwater sound research, espe¬ 
cial^ during peacetime, should be to develop such 
an understanding. 

The present volume presents the essential results 
obtained in the studies of underwater sound up to 
the middle of 1945. This volume was written pri¬ 
marily from the fundamental viewpoint of scientific 
research; in other words, the data are presented 
against a framework of an attempted understanding 
of the factors involved rather than as an unadorned 
summary of the experimental results. Since the meas¬ 
urements were not carried out primarily to increase 
this understanding, this presentation of the subject 
leads to many obvious gaps. However, it is hoped 
that the overall scientific picture presented will be 
stimulating to any future research workers in this 
field. To aid those interested in application, practical 
summaries of the results are given at the end of each 
of the four parts comprising this volume. 

Since our understanding of the details of under¬ 
water sound has not been sufficient in most cases to 
allow an elaborate comparison between theory and 
experiment, it has been possible in most of this volume 
to write the text on the level of a senior engineering 
student. A deliberate effort has been made to keep 
to this level wherever possible in order to make the 
results available to the widest possible group of 
readers. However, more elaborate theoretical devel¬ 
opments have been included where it was believed 
that they were essential to an understanding of the 
full significance of current information. 

The first two parts of this volume deal with the 
propagation of sound in the absence of targets. Part I 
discusses the transmission loss of sound sent out from 
a projector, while Part II deals with sound which has 


IX 


X 


PREFACE 


been scattered back to the vicinity of the original 
sound source. Part III deals with the echoes returned 
from submarines and surface vessels. Part IV dis¬ 
cusses the transmission of sound through wakes and 
echoes received from wakes. 

It should be emphasized that, this work is essen¬ 
tially a report of the work carried out by the Univer¬ 
sity of California Division of War Research in col¬ 
laboration with the U. S. Navy Electronics Labora¬ 
tory, formerly the U. S. Navy Radio and Sound 
Laboratory; and by the Woods Hole Oceanographic 
Institution. Both the Underwater Sound Reference 
Laboratories and the New London Laboratory of 
Columbia University Division of War Research, as 
well as the underwater sound laboratory of the 
Massachusetts Institute of Technology, have also 
made important contributions in special fields. All 
these groups, under contract with Division 6, have 
been very helpful in the preparation of this volume. 
They have at times supplied unpublished data and 
have made many helpful comments and suggestions 
for improving the presentation. 

The direct preparation of this volume has been 
largely a cooperative enterprise of the Sonar Analysis 
Group, operating under different auspices at different 
times. This work was initiated under the Special 
Studies Group of Columbia University Division of 
War Research, under Contract OEMsr-1131. Most 
of the writing was done by the Sonar Analysis Group 


under Contract OEMsr-1483; during this time, the 
Group operated under the auspices of the Office of 
Field Service but under the cognizance of Section 940 
of the Bureau of Ships. Final preparation of the 
manuscript was completed while the Group formed 
part of the Woods Hole Oceanographic Institution 
under Contract Nob.s-2083 with the Bureau of Ships. 

The scientific staff of the Sonar Analysis Group 
engaged in this work were: P. G. Bergmann, E. 
Gerjuoy, P. G. Frank, A. N. Guthrie, Lieut, (jg) 
J. K. Major, USNR (Project Officer of the Group), 
J. J. Markham, L. Spitzer, Jr., R. Wildt, and A. 
Yaspan. The work was under the general supervision 
of the director of the Group, aided by A. N. Guthrie, 
Administrative Director. The editors were: 

Part I P. G. Bergmann and A. Yaspan 
Part II E. Gerjuoy and A. Yaspan 
Part III Lieut, (jg) J. K. Major, USNR 
Part IV R. Wildt 

Because Part I was largely the result of cooperative 
effort by many members of the Group, as well as 
by C. Herring of the Special Studies Group, names of 
individual chapter authors of Part I are listed in the 
Table of Contents. Final assembty of the material 
was under the supervision of Mrs. E. E. Wagner, and 
of H. Birnbaum and M. Ivlapper. 

Lyman Spitzer, Jr. 

Director, Sonar Analysis Group 



CONTENTS 


CHAPTER PAGE 

PART I 

TRANSMISSION 

1 Introduction by P. G. Bergmann and L. Spitzer, Jr. . . 3 

2 Wave Acoustics by P. G. Frank and A. Yaspan .... 8 

3 Ray Acoustics by P. G. Frank, P. G. Bergmann, and A. 

Yaspan .41 

4 Experimental Procedures by P. G. Bergmann .69 

5 Deep-Water Transmission by L. Spitzer, Jr . 86 

6 Shallow-Water Transmission by P. G. Bergmann . . . 137 

7 Intensity Fluctuations by P. G. Bergmann .158 

8 Explosions as Sources of Sound by C. Herring . . . . 173 

9 Transmission of Explosive Sound in the Sea by C. Herring 192 

10 Summary by L. Spitzer, Jr. and P. G. Bergmann . . . 236 

PART II 

REVERBERATION 

11 Introduction.247 

12 Theory of Reverberation Intensity.250 

13 Experimental Procedures.272 

14 Deep-Water Reverberation.281 

15 Shallow-Water Reverberation.308 

16 Variability and Frequency Characteristics.324 

17 Summary.334 


XI 












Xll 


CONTENTS 


CHAPTER PAGE 

PART III 

REFLECTION OF SOUND FROM SUBMARINES 
AND SURFACE VESSELS 

18 Introduction.343 

19 Principles.345 

20 Theory .352 

21 Direct Measurement Techniques.363 

22 Indirect Measurement Techniques.379 

23 Submarine Target Strengths.388 

24 Surface Vessel Target Strengths.422 

25 Summary.434 

PART IV 

ACOUSTIC PROPERTIES OF WAKES 

26 Introduction.441 

27 Formation and Dissolution of Air Bubbles.449 

28 Acoustic Theory of Bubbles.460 

29 Velocity and Temperature Structure.478 

30 Technique of Wake Measurements.484 

31 Wake Geometry.494 

32 Observed Transmission Through Wakes.503 

33 Observations of Wake Echoes.512 

34 Role of Bubbles in Acoustic Wakes.533 

35 Summary.541 

Bibliography.547 

Contract Numbers.557 

Service Project Numbers.558 

Index.559 























PART l 


TRANSMISSION 





Chapter 1 


INTRODUCTION 


l.l IMPORTANCE OF TRANSMISSION 
STUDIES 

S ince sound waves are transmitted through water 
very much more readily than radio and light 
waves, the use of underwater sound has become a 
basic part of subsurface warfare. There are always 
many different ways in which equipment can be de¬ 
signed and used. An intelligent choice between the 
different alternatives depends on accurate knowledge 
of the different factors affecting final performance. 
One of these factors is the extent to which sound is 
weakened in passing from one point to another; this 
weakening is called transmission loss. The present 
volume summarizes the information available in 1945 
on transmission loss of underwater sound. a Much of 
the detailed discussion refers to a sound frequency of 
24 kc since this is the frequency most commonly used 
in practical echo ranging, and most of the available 
data are at that frequency. 

This information, although incomplete, is useful 
in a variety of ways. In particular, it is helpful both 
in the design of gear and in the development of opera¬ 
tional doctrine. 

It is evident that the intelligent design of new 
equipment requires reliable information on under¬ 
water sound transmission as well as on a variety of 
other factors. For example, the choice of frequency 
in any device usually involves a compromise between 
high frequency for the sake of directivity and low 
frequency for the sake of good transmission. It is 
possible to arrive at a suitable compromise by trial- 
and-error methods. However, the choice is made more 
quickly if routine methods can be used to predict the 
transmission loss at each frequency, the directivity, 


a This volume includes primarily those data applicable in 
the frequency range above 200 cycles. Sound of lower fre¬ 
quencies has not been used in sonar equipment and its trans¬ 
mission has not been investigated by Division 6 of the NDRC, 
except occasionally in connection with the transmission of 
explosive pulses. 


and other factors, such as the noise level, which affect 
performance. These different predictions can then 
be combined to find which frequency gives the best 
results. The optimum frequency will, of course, de¬ 
pend on the purpose for which the equipment is de¬ 
signed, and on the limitations of size, available 
power, and other characteristics. Thus, in some types 
of echo-ranging equipment, low-frequency gear with 
a wide beam pattern and a long maximum range is 
used in searching for submarines, but tilting high-fre¬ 
quency gear is provided for tracking a submarine at 
close range during an attack. 

The development of operational doctrine for the 
gear already in use also depends on the results of 
transmission studies largely because of the wide 
variability of underwater sound transmission. If a 
pulse of sound is sent into the water and received near 
the surface 3,000 yd away, the signal energy received 
will sometimes be only a millionth of the signal energy 
received at other times. This enormous variation is 
due mainly to changes in the vertical temperature 
gradients present in the water. These changes have a 
direct effect on the maximum range at which sub¬ 
marines can be detected by echo-ranging gear. When 
the maximum range is known to be short, the gear 
can be operated most effectively with a short keying 
interval, since more rapid keying increases the chances 
of finding a submarine which happens to be within the 
maximum range. If a long keying interval were used 
under these conditions, time would be wasted in 
listening for echoes during periods when no echoes 
would be possible. 

Information on the change of sound transmission 
conditions with changing temperature conditions is 
useful in the choice of antisubmarine tactics as well as 
in the selection of rules for operating the sonar gear. 
When the transmission loss of sound is high and the 
maximum range of sonar gear is short, the spacing 
between surface vessels conducting an antisubmarine 
hunt must be reduced. Sharp temperature gradients 
at considerable depths may weaken sound passing 


3 



4 


INTRODUCTION 


through them, and reduce the maximum range on a 
deep submarine to much less than the maximum 
range on the same submarine at periscope depth. The 
maximum range at which two surface ships can ob¬ 
tain echoes from each other gives, by itself, no infor¬ 
mation on the maximum range that can be expected 
on a deep submarine. Thus, use of the bathythermo¬ 
graph is required to estimate the approximate maxi¬ 
mum range obtainable on a submarine at evasive 
depths. Such an estimate is useful not only in the 
choice of spacing between antisubmarine vessels but 
also in evaluating the desirability of detaching escort 
vessels from a convoy to hunt a submarine reported 
sighted some distance away. When sound conditions 
are good, detaching antisubmarine vessels is less likely 
to endanger the convoy and more likely to sink the 
submarine than when sound conditions are bad. 

Information on sound transmission conditions is 
also useful in the choice of submarine tactics. A sub¬ 
mariner is free to choose his depth of operation, and 
one of the factors influencing this choice is the maxi¬ 
mum range at which he is likely to be detected at 
each depth. In any case, the behavior of the sub¬ 
marine may be influenced by knowledge of the maxi¬ 
mum range at which detection may be expected. At 
times, transmission conditions are so severe that the 
submarine cannot be detected even at 500 yd; such 
conditions, if they can be readily and reliably identi¬ 
fied, provide opportunity for unusually aggressive 
action. 

1.2 NATURE OF SOUND 

Historically, the various types of physical phe¬ 
nomena were first defined in terms of the human 
senses. Physics was divided into the fields of (1) me¬ 
chanics (dealing with touch and displacements ef¬ 
fected by human muscle power), (2) light (dealing 
with the perception of objects by the eye), (3) sound 
(pertaining to hearing), (4) heat (dealing with the 
sensations of heat and cold), and other similar fields. 
Gradually, as the causes of the nerve stimuli became 
understood, the subject matter of physics was re¬ 
grouped; classification in accordance with physiologi¬ 
cal perception was gradually replaced by classifica¬ 
tion according to the physical nature of the phenom¬ 
ena studied. Thus, optics became more and more a 
subdivision of the theory of electricity and magnet¬ 
ism, while heat and sound came to be treated as sub¬ 
divisions of mechanics. The theory of heat is con¬ 
cerned with random motions of many particles. In 


contrast, sound is concerned with the formation and 
propagation of vibrations, primarily in a fluid, b at 
frequencies both within and above the range of audi¬ 
bility. This definition is purely arbitrary, dictated by 
practical considerations, and may be ambiguous 
under certain circumstances. Nevertheless, it is gen¬ 
erally accepted. 

The physics of sound is usually called acoustics. 
Although a major part of the work in acoustics deals 
with sound perceptible by the human ear (the acous¬ 
tics of rooms, the physiology of sound, and similar 
subjects), inaudible sound, consisting of mechanical 
vibrations above the range of frequencies perceived 
by the ear, has come to play an important role in 
subsurface warfare. In this volume on the proper¬ 
ties of sound in the ocean, more than half of the dis¬ 
cussion will be devoted to the propagation of super¬ 
sonic sound, that is, sound at frequencies well above 
those which can be heard. 

1 . 2.1 Sound as Mechanical Energy 

It must be understood that sound energy is a form 
of mechanical energy. The particles of a fluid in 
which sound is traveling are set in motion and tem¬ 
porary stresses are produced which increase and de¬ 
crease during each vibration. The motion of the indi¬ 
vidual particles gives the fluid kinetic energy while 
the stresses induce potential energy. In acoustics, the 
sum of these two kinds of energy is called sound 
energy or acoustic energy. It is not always easy to 
separate the acoustic energy from other forms of 
mechanical energy possessed by the fluid. 

A fluid obtains acoustic energy by some kind of 
energy transformation. As an illustration, consider a 
tuning fork in air. When this tuning fork is struck 
with a rubber hammer, its two prongs are set in 
rhythmic vibratory motion. The vibrating prongs of 
the tuning fork produce compressions and rarefac¬ 
tions in the surrounding air by pushing the adjacent 
air mass away and then permitting it to rush back. 
These alternating compressions and rarefactions are 
propagated through the air and may be detected as 
sound by a suitable instrument, such as the human 
ear or a microphone. The original source of energy 
was the rubber hammer, which had kinetic energy of 
translation. This energy was transformed, by means 
of a collision, to vibratory energy in the tuning fork, 

b The term fluid, as used in physics and chemistry, means 
any liquid or gaseous substance. Thus air and water are fluids, 
but steel is not. 




NATURE OF SOUND 


5 


which was communicated to the air as acoustic 
energy. 

The foregoing example illustrates the general proc¬ 
ess by which sound is generated and detected. A 
source of sound converts mechanical or electrical 
energy into energy of vibration and communicates 
this energy to the surrounding medium as acoustic 
energy. This acoustic energy travels through the 
medium to the receiving instrument where it is de¬ 
tected. 

1.2.2 Production and Reception 
of Sound 

Most types of sonar gear produce sound by con¬ 
verting electrical energy into acoustic energy and de¬ 
tect sound by converting acoustic energy into elec¬ 
trical energy. They do this by making use of one of 
two effects, magnetostriction or the piezoelectric effect. 

When certain metals, such as nickel, are placed in a 
magnetic field, they contract (or expand) in the direc¬ 
tion of the field; conversely, when they are subjected 
to a contracting (or expanding) force they become 
partially magnetized. Thus, if a nickel rod is made 
the core of a solenoid and if it is given a permanent 
magnetization by means of a direct current, then an 
alternating current passed through the winding will 
cause the magnetization to increase and decrease 
with the frequency of the current. As a result, the rod 
will contract and expand or, in other words, vibrate 
with the frequency of the impressed current. In this 
arrangement, electrical energy is converted into 
acoustic energy which is passed into the surrounding 
medium. Conversely, if a sound wave hits this instru¬ 
ment and causes the nickel rod to alternately expand 
and contract, the rod will be magnetized and demag¬ 
netized rhythmically, thus inducing an electromotive 
force in the surrounding solenoid. The resulting alter¬ 
nating current may be amplified and ultimately re¬ 
corded in one form or another. Such a magnetostric¬ 
tion transducer may thus be used both as a source of 
sound and as a receiver of sound. 

Certain crystals, such as quartz, Rochelle salt, and 
ammonium dihydrogen phosphate, exhibit the piezo¬ 
electric effect. If a slice is cut from such a crystal and 
if an electric potential difference is applied across 
such a slice, the crystal will either contract or expand, 
depending on which of the two faces is electrically 
positive. Conversely, if such a slice is compressed or 
expanded mechanically, the two opposite faces will 
develop a potential difference. Thus, a piezoelectric 


crystal, or an array of such crystals, may be used as a 
transducer. If an alternating voltage is applied to the 
opposite sides of the crystal slice, it will vibrate with 
the frequency of the applied voltage; and if it is 
placed in a fluid where the pressure is fluctuating, it 
will develop a fluctuating emf across its faces. 

Other important sources of waterborne sound are 
underwater explosions, ships, submarines, waves, 
underwater ordnance, and biological sources. 

1.2.3 Propagation of Sound 

Chapters 1 through 10 are concerned with the prop¬ 
agation of sound in the ocean. The complexity of this 
problem is due to the great variability of the mechani¬ 
cal properties of the medium in which the propaga¬ 
tion takes place, but the basic underlying physical 
concepts are fairly simple. These principles are dis¬ 
cussed in the following sections. 

Direction of Propagation 

Sound energy is propagated away from the source 
into a medium. If a single pulse of sound is considered, 
such as that produced by a sudden explosion, the 
course of the sound energy in the medium can be fol¬ 
lowed by placing a large number of recording micro¬ 
phones in the general vicinity and by noting the 
times at which they show the first response. Each will 
respond at a slightly different time. Some, placed be¬ 
hind obstructions, may not respond at all. 

By using a sufficiently large number of such micro¬ 
phones, we can record all those points in space which 
are reached by the spreading sound pulse at the same 
time. We shall call the surface on which these points 
are located a sound front (a better expression will be 
introduced later). The progression of the pulse in 
space may then be described by a succession of sound 
fronts along with the statement of the time at which 
each front is activated. If the medium of propagation 
is homogeneous, the perpendicular distance between 
two sound fronts is proportional to the time it takes 
the sound pulse to travel from one to the other. In 
other words, in a homogeneous medium sound travels 
at a constant speed in a direction perpendicular to the 
sound front. This direction is called the direction of 
propagation. 

These simple rules apply only if the sound beam 
meets no obstructions. If an obstruction is placed be¬ 
tween source and microphone, the microphone usu¬ 
ally registers some sound, but with a delay indicating 



6 


INTRODUCTION 


that the sound pulse had to travel “around the cor¬ 
ner” to reach the microphone. In that case, sound 
energy is obviously deflected around the obstruction; 
and it can be shown that this energy does not travel 
everywhere in a direction normal to the sound front. 
A rigorous treatment of these more involved cases 
shows, nevertheless, that a direction of propagation 
always can be defined in a natural and unique man¬ 
ner. 

Intensity of the Sound Field 

Sound is weakened as it travels and at very great 
distances from the sound source cannot be detected. 
We specify the strength of the sound by its intensity. 
Sound intensity is defined as the rate at which sound 
energy passes through an area 1 centimeter square 
placed squarely in the path of the traveling sound. 

In theoretical studies sound intensity is usually 
expressed in units of ergs per square centimeter. In en¬ 
gineering work, on the other hand, it is usually more 
practical to express intensities on a logarithmic scale 
both because of the very wide range of sound inten¬ 
sities in practice and because sound intensities are 
frequently the product of several factors. Use of the 
logarithmic scale narrows down the numerical range 
between very faint and very loud sounds and also 
simplifies the computation of many sound intensities 
by replacing multiplication by addition. 

The logarithmic scale in general use is the decibel 
scale. This scale may be explained as follows. Sup¬ 
pose we want to compare two sound intensities 7i and 
7 2 . To find the decibel difference between A and / 2 , 
the common logarithm (base 10) of the ratio 1\/Ii is 
multiplied by 10. As an example, suppose the inten¬ 
sity 7i is 1,000,000 times the intensity 7 2 . The loga¬ 
rithm of 1,000,000, multiplied by 10, is 60. Thus the 
intensity 7i is 60 db above the intensity 7 2 . In many 
studies it is the decibel difference between two dif¬ 
ferent sounds rather than the absolute strength of 
any one sound, which is of most interest and can be 
most readily determined. 

The decibel scale is also suitable for expressing ab¬ 
solute sound intensities. For this purpose, a standard 
intensity is first selected, called the reference intensity 
or reference level, and then all other sound field in¬ 
tensities are expressed in terms of decibels above (or 
below) the standard. Unfortunately, different stand¬ 
ards have been used by different groups in under¬ 
water sound research. Sometimes, 10~ 16 watt per sq 
cm has been used as the standard since this is the usu¬ 
ally accepted standard in air. More frequently, the 


reference level has been expressed in terms of the 
sound pressure. 

Since sound represents vibrations and since vibra¬ 
tions of a fluid (such as air or sea water) are associated 
with periodic changes in the local pressure, the devia¬ 
tion of instantaneous pressure from the hydrostatic 
or atmospheric pressure may be used as a measure of 
sound intensity. This excess pressure oscillates dur¬ 
ing each cycle; therefore, the intensity must be ex¬ 
pressed in terms of some averaged quantity. Since the 
excess pressure is positive during one half of the cycle 
and negative during the other, its arithmetic mean 
vanishes. It is possible to obtain a nonvanishing 
average quantity by considering the rms excess pres¬ 
sure. In the case of a sinusoidal vibration, the rms 
excess pressure is equal to l/\/2, or 0.7 times the 
maximum value of the excess pressure. It will be 
shown in Chapter 2 that in a given medium the sound 
intensity is proportional to the mean square excess 
pressure. Two standards based on pressure have been 
used in underwater sound studies. One is a sound in¬ 
tensity corresponding to an rms excess pressure of 
0.0002 dyne per sq cm. This standard has been re¬ 
cently replaced by that of an intensity corresponding 
to an rms excess pressure of 1 dyne per sq cm. When 
sound field intensities are expressed on a decibel scale 
relative to some standard intensity, they are usually 
referred to as sound levels. 

1.3 PROPAGATION OF SOUND IN 
THE SEA 

When the propagation of sound in the sea first be¬ 
came a matter of prime military importance, it was 
hoped and expected that sound would travel along 
straight lines from the source and that the sound 
field intensity would decrease in accordance with the 
simple inverse square law. However, this hope was 
not realized. Because of the peculiar characteristics 
of the ocean as a sound-transmitting medium, marked 
deviations occur from both straight-line propagation 
and inverse square intensity decay. 

Straight-line propagation of sound is to be expected 
only if the velocity of propagation is constant 
throughout the medium. In the ocean this condition 
is usually violated primarily because of the variation 
of temperature with depth. There is almost always 
a layer in which the water temperature drops ap¬ 
preciably with increasing depth. This layer may begin 
right at the sea surface, or it may lie beneath a top 
layer of constant temperature. In such a region of 



PROPAGATION OF SOUND IN THE SEA 


7 


temperature change, the sound paths are bent in the 
direction of lower velocity of propagation, in other 
words, in the direction of lower temperature. Even 
though the changes in sound velocity are small (about 
1 per cent for a temperature drop of 10 F), the result¬ 
ant bending of the sound path becomes appreciable 
over a distance of a few hundred yards. If, for in¬ 
stance, the drop in temperature begins directly at the 
surface of the water, and totals a degree or more 
in 30 ft of depth, most of the sound energy will travel 
along paths bent downward and will miss a shallow 
target at a range of 1,000 yd. 

Because of this bending of sound by temperature 
gradients, some departure of sound intensity from the 
inverse square law is to be expected. The amount of 
this departure can be calculated if the temperature 
distribution in the ocean is known. However, even 
this more complicated process for computing the in¬ 
tensity is too simple. The effects of the boundaries of 
the medium (ocean bottom and surface), and of the 
absorption and scattering of sound in the body of 
the ocean must, also, be considered. 

Both the sea surface and the sea bottom affect the 
sound field intensity. Some of the sound energy 
strikes these boundaries and is then partly reflected 
back into the ocean, partly permitted to pass into the 
adjoining medium (air or sea bottom). The portion of 
the energy which is reflected will return into the in¬ 
terior in a variety of directions. Also, little under¬ 


stood processes in the body of the ocean affect sound 
intensity. In some way, a certain amount of the pass¬ 
ing sound energy is converted into heat (absorption 
of sound); and chance impurities such as fish, sea¬ 
weed, plankton, and gas bubbles, tend to scatter a 
small amount of the passing sound energy in all direc¬ 
tions out of its principal path. 

For all these reasons, the propagation of under¬ 
water sound presents, at first, a rather confusing 
picture. Considerable progress has been made, how¬ 
ever, in understanding the behavior of underwater 
sound and in utilizing this partial understanding in 
the design and tactical use of sound gear. The results 
which have been achieved are due to a combination 
of theoretical and experimental investigations. Chap¬ 
ters 2 through 10 discuss the background and progress 
of these investigations. Chapters 2 and 3 lay the 
theoretical groundwork for the physics of underwater 
sound. Chapter 4 leads toward the experimental re¬ 
sults by reporting on the equipment and procedures 
employed in the experiments. Chapters 5 and 6 re¬ 
port experimental results on the propagation of 
sound, primarily sound generated by transducers. 
Chapter 7 is concerned with the observed short-term 
fluctuation of underwater sound intensity. Chapters 
8 and 9 deal with the formation and transmission of 
explosive sound. Finally, Chapter 10 summarizes the 
results obtained to date and discusses possibilities for 
future research. 



Chapter 2 


WAVE ACOUSTICS 


S ound energy takes the form of disturbances of 
the pressure and density of some medium. There¬ 
fore, the basic relationships between impressed forces 
and resulting changes in pressure and density are use¬ 
ful in an understanding of sound transmission. In this 
chapter we shall derive several such relationships, 
and shall combine them into one differential equation 
relating the time derivatives and space derivatives of 
the pressure changes to several physical constants of 
the medium itself. This differential equation is the 
foundation for the mathematical treatment of sound 
transmission to which the rest of the chapter is 
devoted. 

We shall see that this mathematical approach can¬ 
not in itself furnish complete information on sound 
transmission in the ocean. The physical picture must 
necessarily be simplified to make mathematical de¬ 
scription possible — and even this simplified scheme 
does not yield explicit results for the sound intensity 
in all cases. However, it is valuable to know the 
mathematical theories even if they are partially un¬ 
successful in predicting the qualities of sound trans¬ 
mission. Tendencies predicted by a simplified theory 
are often verified qualitatively in practice. Also, 
there is always the hope that by changes and ampli¬ 
fications an incomplete theory can be made much 
more useful. 

2.1 BASIC EQUATIONS 

In this section we shall derive the basic equations 
which will be put together to derive the fundamental 
differential equation of wave propagation, the wave 
equation. These equations are (1) the equation of 
continuity, which is the mathematical expression of 
the law of conservation of mass; (2) the equations of 
motion, which are merely Newton’s second law ap¬ 
plied to the small particles of a disturbed fluid; 
(3) force equations, which relate the fluid pressure 
inside a small volume of the fluid to the external 
forces acting on the periphery of the volume; (4) the 
equation of state, which relates the pressure changes 
inside a fluid to the density changes. 


2.1.1 Equation of Continuity 

The equation of continuity is simply a mathemati¬ 
cal statement of the law that no disturbance of a fluid 
can cause mass to be either created or destroyed. In 
particular, any difference between the amounts of 
fluid entering and leaving a region must be accom¬ 
panied by a corresponding change in the fluid density 
in the region. 

To express this law in mathematical terms we must 
first derive an expression for the mass of fluid which 
passes through a certain small area of a surface in one 
second. Let the small surface element have the area 
A, as in Figure 1, and let the fluid move in a direction 


t*o t*i 


/—*-7 

7 —-A— 

4 

1 

* 

/ 


Figure 1. Passage of fluid through area element A. 

perpendicular to A with the velocity u. In one second, 
a rectangular fluid element of base A and height u has 
passed through this element of area; that is, a volume 
of Aw cubic units of fluid has traversed the area. The 
mass of fluid passing through the area per second will 
thus be pAw, where p is the density at the point and 
time in question. If the fluid is moving not perpen¬ 
dicular to the element A, but in some other direction, 
the mass passing A per second will still be given by 
pA w, if w is taken to be the velocity component in the 
direction perpendicular to A. 

Now consider a small hypothetical box-shaped 
volume inside the fluid, and examine the amounts of 
fluid entering and leaving this box (pictured in 
Figure 2). For simplicity, we can assume that the 
edges of the box are parallel to the coordinate axes. 
Let the dimensions of the box be l x ,l y ,k, as shown in 
the diagram, and let the coordinates of the point H be 
( x,y,z ). Let the components of the fluid velocity at 
the point H be w*,w„,w 2 . 


8 














BASIC EQUATIONS 


9 


The mass of fluid entering the face AHED in unit 
time is clearly the rate at which mass is moving in the 
x direction times the area of AHED, or pu x l u l x . The 
mass of fluid leaving the box through BCGF is a 


A B 



similar expression, but with p and u x measured at 
{x + l x ,y,z). The value of pu x at (x + l x ,y,z) is just 
its value at ( x,y,z ) increased by l x d(pu x )/dx since l x is 
very small. That is, the mass leaving in one second 
through face BCGF is 

-J- 

Then the net increase per second in the mass inside 
the box caused by the flow through the two faces 
perpendicular to the x axis is 

- (pU x )l x lyl z . 

dx 

Similarly, the net increase per second caused by the 
flow through the two faces perpendicular to the y 
axis is 

T~(p u v)lxlylzt 
dy 

and through the two faces perpendicular to the z axis, 
d 
dz 

The total time rate of increase in the mass con¬ 
tained in the box is simply the sum of these three 
quantities, or 


-[ 


d(pU x ) d(pUy) 


+ 


3 (pit 




l x lyl z - 


( 1 ) 


dx dy ' dz 

Since no mass can be created or destroyed inside 
the box, this rate of deposit of mass must result in a 
corresponding change in the average density p inside 
the box. That is, 




"[ 


d(pU x ) d(pUy) d(pU 


+ 


+ 


-]uk 


dx dy dz 

Canceling out the constant factor l x l u l z , we obtain the 
general equation of continuity 

d(pu x ) , d{pu y ) , d(pU z ) 


3p _ _ P 
dt L 


+ 


+ 


a 


( 2 ) 


dx dy dz 

This equation can be simplified if it is assumed 
that all displacements and changes of density are so 
small that second-order and higher products of them 
can be neglected. The actual density p, then, will not 
be very different from the constant equilibrium den¬ 
sity po. If a is defined by 

p — Po 


Po 


( 3 ) 


then a, the fractional change in density caused by the 
displacement of the fluid from equilibrium, will be a 
very small number. Henceforth a will be called the 
fractional density change or condensation. 

With this understanding, it is clear that 

d(pu x ) d r/ , N 
7 = r“l_(po + Po<r)U x J 

dx dx 

d du x 

— T~\PoU x ) — Pot , 
dx dx 


since the second-order product <ru x can be neglected. 
By substituting this value of d(pu x )/dx and similar 
expressions for d(pu v )/dy and d(pu x )/dz into equation 
(2), the following simplified equation of continuity 
results: 

da (du x du y du\ 

«““Vte + a y + to)' (4) 

Equation (4) is the form of the equation of con¬ 
tinuity which will be used in the derivation of the 
wave equation (27). 

For later reference, we shall note what happens to 
the volume occupied by an infinitesimal mass of the 
fluid when the fluid is given a small displacement 
from equilibrium. If v 0 is the volume occupied by the 
small mass at equilibrium, and v is the volume at 
time t, then a fractional volume change a can be de¬ 
fined by 


v — v 0 

CO = - 

Vo 


( 5 ) 























10 


WAVE ACOUSTICS 


From equations (3) and (5), 

P = p 0 (l + a), (6) 

v = t» 0 (l + w). (7) 

Since the masses at equilibrium and at time t are 
equal, 

pv = p 0 v 0 . (8) 

By combining equations (6), (7), and (8) 

(l + «)(1 + < 0 * 1 . 

The product ua, a second-order term, can be neg¬ 
lected, giving 

w = — cr. (9) 

That is, under the assumption of small displacements 
and small density changes, the fractional volume 
change w is the negative of the fractional density 
change <r. 


2.1.2 Equations of Motion 

In this section, we shall apply Newton’s second law 
of motion to the mass of fluid within the volume ele¬ 
ment d 0 . This law states that the product of the mass 
of a particle by its acceleration in any direction is 
equal to the force acting on the particle in that direc¬ 
tion. 

Given the velocity distribution within the fluid as 
a function of the position coordinates and time, 

u z = u x (x,y,z,t), etc.; (10) 

then the distribution of acceleration within the fluid 
is to be calculated, 

a z = a z {x,y,z,t), etc. (11) 

We cannot immediately say that a z = du z /dt. For, 
in order to calculate the acceleration at a particular 
point and a particular time, we must focus attention 
on one particular particle. At the end of a time incre¬ 
ment dt, the particle has moved to a point (x + dx, 
y -f dy,z + dz), where it has the velocity component 
u z (x + dx, y -f dy, z dz). The difference between 
its new velocity and its original velocity, divided by 
the time interval dt, gives the desired acceleration 
component du z /dt. This value is not exactly the same 
as the simple partial derivative du z /dt, because the 
latter does not focus attention on the change of 
velocity of a single particle, but instead compares the 
velocity of a particle at the point ( x,y,z ) and time t 
with the velocity of the particle which occupies the 
position ( x,y,z) at the end of the time interval dt. 
However, du z /dt and du z /dt will be almost equal 


under the assumptions that second-order products 
of displacements, particle velocities, and pressure 
changes are negligible. To show this, we note that 


du z 

dt 


du z du z dx 
dt dx dt 


du z dy du x dz 
dy dt dz dt ’ 


( 12 ) 


which is the usual equation found in calculus texts re¬ 
lating the partial and total time derivatives of a 
function. The last three terms are second-order prod¬ 
ucts, since dx/dt, dy/dt, dz/dt are merely u x ,u v ,u z . 
Thus, the component of acceleration in the x direc¬ 
tion may be approximated by du z /dt, and similarly 
for a v and a z . That is, 


a z 


du z 
dt ’ 


CLy 


dliy 

dt ’ 


a. 


du t 

dt 


(13) 


The mass of fluid within the volume element v 0 is 
pv 0 . If F Z ,F V ,F Z are the components of the forces acting 
on the element, then the equations governing the 
motion of this element are, in view of equation (13), 


du z du u 

F z pva , Fy pv o 

ut ot 


dll. 

Fz = pv o—• (14) 

Ot 


It is desirable to make the equations governing the 
motion of the small element independent of the par¬ 
ticular value of the small volume v 0 . For this reason, 
we rewrite equations (14) as 



where 


du u 
fu ~ P dt 




Ft 
Vo ’ 



fz = P 


du z 
dt ' 



(15) 


The normalized force components may be regarded 
as the force components per unit volume acting on the 
small volume element. 

The next section is concerned with calculating 
f z ,f v ,fz in terms of the pressure or density changes oc¬ 
curring within the fluid. 


2.1.3 Law of Forces in a Perfect Fluid 

A fluid is called perfect if the forces in its interior 
are solely forces of compression and expansion, in 
other words, if the fluid is incapable of shear stress. 
If a fluid is perfect, the force on any portion of its 
surface is perpendicular to the surface. Fluids which 
can exhibit shear stress in response to shear deforma¬ 
tion, in addition to responding to compressive and 
expansive forces, are called viscous. 



BASIC EQUATIONS 


11 


Before the equations of motion (15) can be used, 
expressions must be derived for the force components 
f z , fv> fz acting on the small box of Figure 2. Accord¬ 
ingly, we shall calculate these forces under the as¬ 
sumption that the fluid is perfect. According to this 
assumption, the box will move in the x direction if 
and only if the pressure on face ADEN is different 
from the pressure on face BCFG. Similarly, it will 
move in the z direction only if the pressures on faces 
ABCD and EFGH are unequal. Motion in the y 
(vertical) direction is not quite so simple because of 
the hydrostatic, or gravity-produced pressure dif¬ 
ferences, which do not of themselves cause motion. 
The box will move in the y direction if and only if 
the pressure on face DCFE is not exactly equal to the 
pressure on face ABGH plus the total weight of the 
box. If the corrected pressure p(x,y,z,t ) is defined as 
the total pressure P(x,y,z,t ) minus the hydrostatic 
pressure at the point ( x,y,z) when the fluid is at 
equilibrium, then the criterion for motion in the y 
direction may be restated as follows. Motion will oc¬ 
cur in the y direction if and only if the corrected pres¬ 
sure at face ABGH differs from the corrected pressure 
at face DCFE. We shall have little occasion to use the 
total pressure P since the hydrostatic pressures are 
seldom important in sound propagation. 



Figure 3. Pressure on opposite faces of infinitesimal 
fluid element. 

Figure 3 is a duplication of the box of Figure 2 
showing the forces acting in the x direction. If the 
pressure at the left-hand surface is p, the total force 
on that surface is pl v l z . The pressure at the right-hand 
surface is clearly p + ( dp/dx)l z ; and the total force 
on that surface is therefore [p + {dp/dx)~\l z l y l z . Since 
the fluid is assumed to be perfect, these forces are 


parallel and their resultant can be obtained by simple 
subtraction. Thus, the total force on the volume v 0 in 
the x direction is given by 

Fz = fxi>o = — ~l x l u lz- (16) 

dx 

Thus, the force per unit volume in the x direction f z 


is given by 

, - 

1 dx' 

(16a) 

Similarly, 

_ dp 

Jy ~ ~ 

dy 

(16b) 


/. - -?• 

(16c) 


From equations (15) and (16a, b, c) we obtain 


dp 

dx 


du z 

dp _ 

dUy 

dp _ 

du z 

’ dt ’ 

dy 

~ P dt ’ 

dz 

~ P dt 


2.1.4 Equation of State 

Our aim is to derive a differential equation which 
will relate certain properties of the disturbed fluid 
(pressure changes, density changes) to the independ¬ 
ent variables x,y,z,t. For effective use, this differen¬ 
tial equation should contain only one dependent 
variable. The basic equations derived up to this point 
— (4), (15), and (16) — contain the dependent vari¬ 
ables cr, p, p, u x , u v , u z . a and p are one variable since 
they are related by equation (3). It will be seen in 
Section 2.2.1 that the velocity components can be 
easily eliminated by use of the equation of con¬ 
tinuity (4). However, we must consider the relation¬ 
ship between density and pressure before we obtain 
a differential equation for the propagation of sound. 
Such a relationship between density and pressure is 
obtained from the equation of state of the fluid. 

The equation of state of any fluid a is that equation 
which describes the pressure of the fluid as a function 
of its density and its temperature, 

P = P(p,T). 

This function P(p,T) must be determined experi¬ 
mentally for each fluid separately. In the case of sea 
water, it depends on the percentage of dissolved salts. 

A relation between pressure and density is obtained 
from the equation of state by making two assump¬ 
tions. First, it is assumed that a passing sound wave 


a We shall follow the common usage of physicists and use 
the term fluids to denote both liquids and gases. 














12 


WAVE ACOUSTICS 


causes the fluid to deviate so slightly from its state 
of equilibrium that the change in pressure is propor¬ 
tional to the fractional change in density. Second, it 
is assumed that the changes caused by the passing 
of the sound wave take place so rapidly that there is 
practically no conduction of heat. We shall denote 
the fractional change in density as heretofore by <r; 
the change in pressure will be called excess pressure 
and denoted by p. 

Thus we assume that the fractional change in 
density and the excess pressure caused by the passing 
sound wave are both small and that they are propor¬ 
tional to each other: 

p = K<J. (18) 

The constant of proportionality k is called the bulk 
modulus. It depends not only on the chemical nature 
of the fluid (such as the concentration of salts in the 
sea), but also on the equilibrium temperature T, 
the equilibrium pressure P 0 , and the equilibrium 
density p. 

The temperature in the ocean varies from point to 
point, usually decreasing with increasing depth. The 
equilibrium pressure increases rapidly with the depth, 
and the density increases very slightly with depth. 
As a result, the bulk modulus k is itself a function of 
all three coordinates x, y, and z, although its greatest 
changes take place in a vertical direction. To the ex¬ 
tent that the temperature distribution of the ocean is 
subject to diurnal and seasonal changes, k is also a 
function of the time t. However, in the following 
sections we shall usually simplify matters by disre¬ 
garding these variations in space and time and by 
treating k as a constant. 


2.2 WAVE EQUATION IN A PERFECT 
FLUID 

2.2.1 Derivation 

If in a certain region of a fluid in equilibrium the 
pressure is changed from its equilibrium value, the 
fluid immediately produces forces which aim toward 
restoring the equilibrium state. Vibrations result, 
which are propagated as waves through the fluid. 
These waves are sound waves, and the fundamental 
differential equation governing their propagation will 
now be developed by using the basic equations de¬ 
rived in the preceding sections. The particular equa¬ 
tions used are the equation of continuity (4), the 


equations of motion (15), the law of forces (16), and 
the equation of state (18). 

From equation (18) we have 

dp da dp da dp da 

dx dx ’ dy dy ’ dz dz 

By putting these values for da/dx, da/dy, and 
da/dz in the law of forces (16) we obtain 


/* = 





(19) 


After these values for the components of the force 
on a small box are substituted into the equations of 
motion (15), we obtain the following relations. 




du v 





( 20 ) 


Since we assume that density changes and velocity 
changes are all comparatively small, the expressions 
pdujdt and p 0 du x /dt will differ by a second-order 
term, and hence can be regarded as equal. Then 
equations (20) become 

du x da 

Po— + k — = 0 
dt dx 


Po' 


du u 

dt 


da 

+ K~ = 0 

dy 

du z da 

Po— + =0. 

dt dz 


( 21 ) 


In order to apply the equation of continuity (4), we 
differentiate the first equation of (21) with respect to 
x, the second with respect to y, and the third with 
respect to z. The equations are added, leading to 

d (du x du v du\ (d 2 a d 2 a d 2 a\ 

*V<W + Ity + aT/ + “w + + W " a 

( 22 ) 

From the equation of continuity, the first parenthesis 
is —da/dt; and equation (22) reduces to 


d 2 a k (d 2 a d 2 a d 2 a\ 

TT = — + — )• (23) 

dt 2 p 0 \dx 2 dy 2 dz 2 / 

, w 
})■ 


PoWX* dy 2 
Since a = p/k, from equation (18), where k is con¬ 
stant, equation (23) implies 

d 2 p k / d 2 p d 2 p dh 

dt 2 po V dx 2 dy 2 dz 2 



WAVE EQUATION IN A PERFECT FLUID 


13 


It can be shown that the velocity components 
u x ,u y ,u z also satisfy a differential equation of the 
form of (23), provided the motion of the disturbed 
fluid is irrotational. That is, if sound is propagated in 
a perfect fluid in such a manner that no eddies are 
produced, 


d 2 U x 

~dt 2 


k I d~U x d 2 U x d 2 U x \ 
Po\dx 2 dy 2 dz 2 ) 


(25) 


with similar equations for u y and u z . 

Equations (23), (24), and (25) are equivalent; that 
is, the fundamental laws of sound propagation can be 
deduced from any one of them by using the known 
relationship between a, u x , and p. In the following 
sections, the most frequent reference is to equation 
(24). Variation in the excess pressure is the most 
familiar and probably the most intuitive change in 
the disturbed fluid; also, the majority of hydrophones 
used in the reception of underwater sound respond 
directly to variations in excess pressure rather than 
to variations in particle velocity or condensation. 

It is convenient, in equations (23) to (25), to set 


c 2 = 


K 

) 

Po 


(26) 


so that the wave equation becomes 

&V = J&V , &7P , . 

dt 2 ° Vdz 2 dy 2 dz 2 / 


(27) 


It will be pointed out in Section 2.3.1 that c, defined 
by equation (26), has the general significance of sound 
velocity. 

The wave equation (27) gives the relationship be¬ 
tween the time derivatives and the space derivatives 
of the pressure in the fluid through which the sound 
is passing. Relationships of this sort have been used 
by generations of mathematicians as a starting point 
for the development of physical theory. In the field 
of sound, these mathematicians have explored the 
methods by which the future course of pressure in a 
fluid can be calculated if only the initial distribution 
of pressure is given. Mathematically, this amounts to 
solving the wave equation (27) with given initial and 
boundary conditions. Once the distribution of pres¬ 
sure is known, the sound intensity at any point and 
time can be calculated by the methods of acoustics. 


2.2.2 Initial and Boundary 
Conditions 

The differential equation of a physical process gives 
a dynamical description of the process relating the 


various temporal and spatial rates of change, but does 
not of itself tell all we want to know. In the case of 
the wave equation, we desire knowledge of how the ex¬ 
cess pressure varies in space and time. This informa¬ 
tion is obtainable, not from the wave equation itself, 
but from its mathematical solution. The general solu¬ 
tion of a partial differential equation like the wave 
equation always contains arbitrary constants and 
even arbitrary functions. These arbitrary constants 
and functions are, in any individual problem, ad¬ 
justed to make the solution fit the special condi¬ 
tions of the problem. 

These special conditions are of two kinds: boundary 
conditions and initial conditions. In the problem of 
sound propagation, the two types of conditions can 
be defined as follows. Boundary conditions are fixed 
by the geometry of the medium itself. If the medium 
is finite, boundary conditions must always be con¬ 
sidered. The excess pressure must fulfill certain con¬ 
ditions at a boundary such as the sea surface, sea 
bottom, or internal obstacle. The pressure may have 
to be zero at one boundary, or a maximum at some 
other boundary, or may satisfy some other condi¬ 
tion. 15 

Initial conditions are concerned not with the fixed 
geometry of the fluid and its surroundings, but with 
the special disturbances which cause sound to be 
propagated. One type of initial conditions specifies 
the pressure distribution at a certain instant of time, 
t = to, over the whole fluid. That is, we are given a 
function p(x,y,z ), and are told that 

p(x,y,z,t 0 ) = p{x,y,z). (28) 

Another type of initial conditions specifies the pres¬ 
sure as a function of time at a fixed point (x 0 ,yo,z o) 
of the fluid. That is, we are given a function p{t) and 
are told that 

p(xo,y 0 ,z 0 ,t) = p{t). (29) 

Every actual case of sound propagation involves 
both initial conditions and boundary conditions. 
However, in our mathematical approximations to 
reality it is best to start with the simplest case, that 
is, where sound is propagated into a medium that is 
infinite. Of course, in theory every problem can be 
regarded as a problem in an infinite medium. We can 
consider the sea and air together as one medium, 
whose physical properties at equilibrium (elasticity, 


b It will be seen in Section 2.6.1 that the excess pressure 
must be nearly zero at the boundary separating water and air 
and a maximum at the solid bottom of the sea. 




14 


WAVE ACOUSTICS 


density, and other properties) suffer a sharp change 
at the separating surface. However, the mathematical 
treatment of a medium with strongly variable physi¬ 
cal properties is still more difficult than the treatment 
of boundary conditions. We are free to schematize 
the physical situation in the most convenient way. 
Accordingly, we shall start with the consideration of 
an infinite medium, where the elasticity and density 
at equilibrium are not strongly variable, and shall 
later specialize our treatment by the consideration of 
boundary conditions (Sections 2.6 and 2.7). 

Initial conditions must always be considered since 
without them no sound could possibly originate. Un¬ 
less the initial conditions are themselves very simple, 
the solution of problems even in an infinite medium is 
quite involved mathematically. The most practical 
procedure is to first find solutions under very simple 
initial conditions and use these solutions as building 
blocks for constructing solutions of problems with 
more complicated initial conditions. 

2.3 SIMPLE TYPES OF SOUND WAVES 

2.3.1 Plane Waves 

It is convenient to start our study of the solution 
of the wave equation with the assumption that the 
disturbance is propagated in layers. We assume that 
at any time t the excess pressure p is a function of x 
only; that is, p is independent of y and z. With this 
understanding, the wave equation (27) reduces to 


The eighteenth century mathematicians knew that 
if p is an arbitrary function of either ( t — x/c) or 
{t x/c), or a sum of two such functions, then p 
satisfies equation (30). The proof is easy. Assume 
that p = fit, — x/c) where/ is any function. Then, 0 



The proof is identical for a function of the argument 
(t + x/c). The sum of two solutions will itself be a 
solution because of the general theorem that the sum 
of two solutions of a homogeneous linear differential 
equation will also satisfy the equation. 


c In accordance with usual calculus notations, f"{t — x/c) 
represents the second derivative of f(z) evaluated for z = 
t — x/c. 


Also, it can be shown that any function of x and t 
which is not of the form 

+ (31) 

cannot possibly satisfy equation (30). The proof is 
carried out as follows. Represent the unknown solu¬ 
tion of equation (30) in the form 

V - f(£,v), £ = x - ct, y = x + ct. 

If the differential equation (30) is written in terms of 
the new variables £ and y, it reduces to 

d'-/(£,7?) = 0 

d£dy 

This equation implies that the first derivative d//d£ 
must be a function of £ only and independent of y, for 
otherwise the second derivative d 2 //d£dr? could not 
vanish. Thus / itself must have the form 

m,v) = m +/*(>?). 

Let us focus attention on all the solutions of equa¬ 
tion (30) which have the form 



There are an infinite number of such solutions cor¬ 
responding to the infinite number of possible choices 
of /. However, no more than one of these solutions 
can fit the special conditions of a particular physical 
situation since the actual pressure at a specified point 
and specified time can have only one value. Suppose 
that there is one member of the family of functions, 
denoted by fft — x/c), which satisfies the given 
initial and boundary conditions. Then a fixed value 
of ( t — x/c), say 4.13, will always be associated with 
some fixed value of the excess pressure, given by 
/i(4.13). If /i(4.13) is equal to 0.02, then the excess 
pressure will be 0.02 at those combinations of time 
and place where ( t — x/c) = 4.13, that is, where 

x = ct — 4.13c. 

In other words, as the time increases, any fixed value 
of the excess pressure travels in the positive x direc¬ 
tion with the speed c. This result is clearly true no 
matter what the form of/i or the particular value of 
the excess pressure. Such a process, in which a given 
pressure change travels outward through a medium, 
is referred to as propagation of progressive waves. 

Similarly, if a function f 2 (t + x/c) is the sole mem¬ 
ber of the family of functions (31) which satisfies the 
imposed initial and boundary conditions, progressive 
waves will be propagated with the speed c in the 





SIMPLE TYPES OF SOUND WAVES 


15 


negative x direction. If, however, an expression of the 
form (31) is the function describing the given physical 
situation, the situation is more complicated. The re¬ 
sulting distribution of pressure will be the mathe¬ 
matical resultant of the pressure distributions calcu¬ 
lated for fi and / 2 ; and a given value of the excess 
pressure will no longer be propagated in a single 
direction with the speed c. Discussion of this more 
complicated type of wave propagation will be de¬ 
ferred until Section 2.7. 

Now we consider two specific examples of the prop¬ 
agation of plane waves in an infinite homogeneous 
medium (no boundary conditions). In the first ex¬ 
ample, we assume as an initial condition that the 
exact pressure distribution is specified at the time 
instant t = 0 between the planes x = 0 and x = x 0 , 
by p(a;,0) = p(x) ; and also that the excess pressure 
is zero at t = 0 for all values of x less than 0 and 
greater than x 0 . We assume that this initial disturb¬ 
ance gives rise to progressive waves traveling in the 
positive x direction, that is, that the solution is of the 
form p = f{t — x/c). Then the solution of the wave 
equation with these conditions must be 

p{x,t) = p(x - ct) (32) 

since first, it satisfies the initial conditions p(x, 0) = 
p{x); second, it is a function of ( t — x/c) and there¬ 
fore satisfies the wave equation (30); and third, there 
can be only one solution to this physical problem. 

By the results of preceding paragraphs, we know 
that a given value of the excess pressure will be 
propagated in the x direction with the speed c. Thus, 
at the time t the initial disturbance will be duplicated 
between the planes x = ct and x = x 0 + ct) and the 
excess pressure will be zero for x < ct and x > x 0 + 
ct. The disturbance of the fluid remains of width x 0 , 
remains unchanged in “shape,” and is propagated 
with the speed c. 

As another example, we suppose as an initial con¬ 
dition that the values of the excess pressure are 
specified only for the plane x = 0, but for the total 
time interval between t = 0 and l — to, by the equa¬ 
tion p(0,t) = p(t) and that the excess pressure at the 
plane x = 0 is zero for t < 0 and t > t 0 . Here, also, 
we assume that this disturbance causes progressive 
waves to be propagated in the positive x direction. 
Arguing as in the preceding example, the solution of 
the wave equation (30) with these imposed conditions 
is 


The expression (33) differs somewhat in form from 
equation (32) because the initial conditions are ex¬ 
tended in time instead of in space. 

In this example, it is known that the excess pres¬ 
sure will be the same at all combinations of space and 
time where x — ct = constant. Since x — ct equals 
zero when x = 0, t = 0, the value of the excess pres¬ 
sure corresponding to x — 0, t = 0, must be assumed 
by the plane x = ct at the time t. Further, since 
x — ct equals — ct 0 at t = to, x = 0, the value of the 
excess pressure corresponding to t = to, x = 0, will 
be assumed by the plane x = ct — ct Q at the time 
instant t. Thus, at time t the disturbance is confined 
between the planes x = ct — d 0 and x = ct; that is, 
the region of disturbance is always of width do and 
is propagated along the positive x axis with the 
speed c. 

Sound Velocity 

We have seen that in some simple situations the 
quantity c may be regarded as the velocity with 
which the disturbance is propagated in the medium, 
or more simply, the velocity of sound in the medium. 
It will be recalled that c was defined in equation (26) 

by 



where k is the bulk modulus and p 0 is the density of 
the fluid at equilibrium. 

If the medium is a perfect gas, the relation of the 
sound velocity to the temperature and pressure can 
be expressed in a simple formula. The pressure 
changes produced by sound in a fluid are usually so 
rapid that they are accomplished without appreciable 
heat transfer, that is, they are practically adiabatic. 
For a perfect gas suffering adiabatic pressure changes 
the bulk modulus is yP, where P is the total pressure 
and y is the ratio between the specific heat at con¬ 
stant pressure and the specific heat at constant 
volume. For a perfect gas suffering any kind of pres¬ 
sure change P = pRT. Thus, the simple result fol¬ 
lows that 

c = VV RT- 

Hence, in air at normal pressure, which is not far from 
a perfect gas, the sound velocity increases with the 
square root of the absolute temperature. 

No such relationship can be derived for the velocity 
of sound in sea water since the pressure, density, and 
temperature of sea water are not related by any 



WATER TEMPERATURE IN DEGREES 


16 


WAVE ACOUSTICS 



DEPTH IN FEET 



Figure 4. Speed of sound in sea water. 



TEMPERATURE IN DEGREES F 
A EFFECT OF TEMPERATURE 



0 10 20 30 40 

SALINITY IN PARTS PER THOUSANO 
B EFFECT OF SALINITY 



DEPTH OF SEA IN FEET 
C EFFECT OF DEPTH 

Figure 5. Percentage variation of sound velocity with 
water temperature, salinity, and depth. A. Effect of 
temperature. B. Effect of salinity. C. Effect of depth. 


simple formula. However, tables have been con¬ 
structed which show the velocity of sound as a func¬ 
tion of three variables which can be measured di¬ 
rectly: the water temperature, pressure, and salin¬ 
ity. Although the relationship is not simple, these 
three variables determine precisely both the bulk 
modulus and the density, from which the sound 
velocity can be calculated from equation (26). 

At 32 F, atmospheric pressure, and normal salinity 
(34 parts per thousand by weight), the velocity of 
sound in sea water is about 4,740 ft per sec. Increase 
of either temperature, pressure, or salinity causes the 
sound velocity to increase. The increase of sound 
velocity with temperature is about 8.5 ft per sec per 
degree F at 32 F, and about 4.0 ft per sec per degree F 
at 90 F. The increase of sound velocity with water 
depth, caused by the increase in pressure, is 1.82 ft 
per sec per 100 ft of depth. In the open ocean for the 
depths of interest in sonar operations the water 
temperature is the controlling factor in determining 
the velocity; since sonar gear is usually operated at 





















































































































































































SIMPLE TYPES OF SOUND WAVES 


17 


shallow depths, the pressure changes are relatively 
unimportant, and salinity changes in the open ocean 
are usually too small to matter much. Near the 
mouths of large rivers, however, where fresh water is 
continuously mixing with ocean water, the variations 
in sound velocity may be largely controlled by varia¬ 
tions in salinity. 

The quantitative dependence of sound velocity on 
temperature, pressure, and salinity is summarized in 
Figures 4 and 5. In Figure 4, obtained from a report 
by Woods Hole Oceanographic Institution [WHOI] 1 
the value of the sound velocity at zero depth can be 
read from the main charts for any given combination 
of temperature and salinity. This velocity can then be 
corrected to the velocity at the actual depth by use 
of the curve in the small box. Figure 5 gives the per¬ 
centage changes in sound velocity caused by specified 
absolute changes in the three determining variables. 
It will be shown in Chapter 3 that it is the relative 
changes in sound velocity which determine whether 
sound transmission is expected to be good or bad 
rather than the absolute changes. 

The direct measurement of sound velocity in the 
ocean is very difficult. The intuitive method of di¬ 
viding distance traveled by time is difficult since 
sufficiently accurate measurement of distances at sea 
is usually not feasible. The U. S. Navy Electronics 
Laboratory at San Diego, formerly the U. S. Navy 
Radio and Sound Laboratory [USNRSL], developed 
an acoustic interferometer for the determination of 
the wavelength of sound at a point in the ocean; 2 
multiplication of this local wavelength by the known 
frequency gives the local sound velocity. This instru¬ 
ment was developed mainly for the purpose of check¬ 
ing whether the temperature changes indicated by 
the bathythermograph were correlated with the actual 
changes of sound velocity in the ocean. Good general 
agreement was observed between the velocity-depth 
plots obtained with the interferometer or velocity 
meter and those computed from bathythermograph 
observations. However, since the bathythermograph 
cannot follow rapid changes in water temperature with 
the detail possible with the velocity meter, a velocity 
microstructure was frequently recorded with the 
meter which deviated as much as 0.1 per cent from 
the velocity calculated from the simultaneous bathy¬ 
thermograph reading. That these deviations were due 
to the slow response of the bathythermograph rather 
than to physical factors was verified by correlating 
the velocity microstructure with the temperature 
microstructure obtained by a thermocouple recorder. 


2.3.2 Harmonic Waves 

Up to now we have allowed the initial disturbances 
p(x,y,z) or p(t) to be arbitrary functions. However, 
most initial disturbances which occur in practice are 
of a very special type that originate in the elastic 
vibration of some medium. Such disturbances are pro¬ 
duced by small displacements of some parts of the 
medium from their positions of equilibrium; these 
displacements in turn produce restoring forces which 
tend to restore the state of equilibrium. Such restor¬ 
ing forces are, in first approximation, proportional to 
the displacements. 

It is well known that under such conditions (re¬ 
storing forces proportional to displacements) the 
initial disturbance must be of the form of a harmonic 
vibration; that is, it must be representable by trigo¬ 
nometric functions of the time. In acoustics, such a 
vibration produces a pure tone of a definite frequency. 
Since echo-ranging pulses are very nearly pure tones, 
the importance of a study of harmonic vibrations is 
obvious. Also, harmonic vibrations are of crucial im¬ 
portance because they are the most convenient build¬ 
ing stones of the more general solutions of the wave 
equation (see Section 2.7). 

Suppose, in the second example under plane waves, 
that the initial disturbance of the plane x = 0 is a 
harmonic vibration. That is, 

p(t ) = a cos 2irf(t — e) 

for values of t between 0 and t 0 . One solution of the 
plane wave equation (30) under initial conditions 
p(Q,t) = p(t) is always 

p(x,t) = p(t ± 

since p(t ± x/c) satisfies the wave equation and also 
the imposed initial conditions. Thus, if we restrict our 
attention to progressive waves traveling in a single 
direction, the solution of the wave equation with 
the given initial conditions is 

p = a cos 2wf(t + - — y 
or 

p = a cos 2 t rf(t — ~ ' (34) 

Clearly, the pressure changes represented by equa¬ 
tion (34) are at most a; for that reason, a is called the 
amplitude of the disturbance. Also, it is clear that at 
a fixed point of space, p goes through / periods in one 
second; and so / is called the frequency of the dis- 



18 


WAVE ACOUSTICS 


turbance. The quantity e is called the phase constant 
because it fixes the position of the disturbance in 
time. Two vibrations of the same frequency and the 
same e have their zeros simultaneously, also, their 
maxima. If they have different e’s, one has its zeros a 
fixed time interval ahead of the other, and we say 
that there is a phase difference between the vibrations. 


2.3.3 Spherical Waves 


The sound at large distances from an actual source 
resembles the sound from a point source more closely 
than it does the sound from an infinite plane. Hence, 
for some purposes it is more realistic to abandon the 
assumption of plane waves, and assume instead a 
point source at the origin which causes the pressure 
in the surrounding medium to be a function only of 
the distance r from the origin and of the time t. That 
is, the pressure is given by some function 

P = P(M) (35) 

and is thus independent of the direction of the line 
joining the source to the point in question. 

We shall now show that the wave equation (27) re¬ 
duces, for the assumed case of spherical symmetry, to 
the simple form (37). 

From simple analytic geometry, 


r 2 = x 2 + y 2 + z 2 ; 


dr 

dx 


x dr 
r dy 


V dr _ z 
r dz r 


(36) 


In order to transform the wave equation, the vari¬ 
ables x,y,z must be eliminated, and the variable r in¬ 
serted. In order to do this, we must use the relations 
(36) to calculate d 2 p/dx 2 , d 2 p/dy 2 , and d 2 p/dz 2 in 
terms of r and the derivatives of p with respect to r. 
This is done as follows. 


dp dp dr dpx 
dx ~ dr dx~ dr r 


because spatially p depends only on r. By differen¬ 
tiating again, 


d 2 p ri + x ' ~\ *!e\ 

~dx 2 ~ dxLdrrJ “ drL r 3 J + rdx\dr) 

m dpi" r 2 - X 2 ] x?drp 

drL r 3 J r 2 dr 2 

Similarly, we can show that 

& V = dp|~ r 2 - y 2 ~ 1 y 2 cPp 

dy 2 drL r 3 J + r 2 dr 2 

d 2 p dpi”r 2 — z 2 ”| z 2 d 2 p 

dz 2 drL r 3 J r 2 dr 2 


Addition of these expressions for drp/dx 2 , d 2 p/dy 2 , 
and d 2 p/dz 2 , in order to obtain the right-hand side 
of equation (27), gives 

*p + *£ + *B = *£ + 2 *E. 

dx 2 dy 2 dz 2 dr 2 r dr 

The latter expression is easily verified to be 
(l/r)(d 2 /dr 2 )(rp), so we finally obtain 


dx 2 dy 2 dz 2 


Id 2 
rdr 2 


( rp ), 


and the general wave equation (27) reduces to 


d 2 (rp) 
dt 2 


dr 2 


(37) 


This equation has a form similar to that of equa¬ 
tion (30) for plane waves with p replaced by rp, and 
x by r. By using an argument similar to that in 
Section 2.3.1, it can be shown that equation (37) is 
satisfied by 



where / is an arbitrary function of one variable. By 
dividing out the r, we get the following expression for 
p(r,t) as the general solution of equation (37): 


p(r,f) = 


f (‘ ~ c) +/ ’(‘ + c) 


(38) 


Assume that the following initial conditions are 
given. The initial disturbance is confined to a spheri¬ 
cal shell of infinitesimal thickness at a distance r = r 0 
from the origin. We suppose that the excess pressure 
in this spherical shell source is given by 

p(ro,t ) = ^ (39) 

r 0 

between the times t = 0 and l = t 0 . We also suppose 
that the excess pressure at points outside this shell 
is zero at the time t = 0. The general solution of 
equation (37) with these initial conditions is 

+ (1 - ci)p (t + ~ c - j)] , (40) 

because first, the right-hand expression is in the form 
of equation (38), and therefore satisfies equation (37); 
second, the right-hand expression satisfies the initial 
conditions imposed. In particular, if the spherical- 
shell source has a very small radius so that it approxi- 










PROPERTIES OF SOUND WAVES 


19 


mates a point source at the origin, the following 
solution is obtained. 


p(r,t) = ci- 


p{‘ + i) 


+ (1 — Ci) 


(41) 


Physically, we can eliminate the solution 
(1/r) p(t-\-r/c). This solution corresponds to wave 
propagation in the negative r direction, with the 
speed c; in other words, to a wave which starts out 
at some negative time with a great radius and con¬ 
tracts into the point x = y = z = Oat the time t = 0. 
The first solution is physically valid since it resembles 
actual propagation from a point source. It implies 
that the spherical wave spreads out from the point 
source into ever-increasing spheres with the speed c. 
Therefore, an initial ping of duration r seconds will 
cause the resulting sound energy to be contained 
within an expanding spherical shell of thickness cr. 

If the source is harmonic (emits a pure tone of the 
frequency/), the initial conditions are of the form 
a cos 2wf(t — e) 


and if r 0 is nearly zero, the pressure at the distance r 
from the source and time t is given by 


P = 


a cos 2irf 


( 



r 


(42) 


The constants / and e have the same physical signifi¬ 
cance as for the plane wave case;/is the frequency of 
the vibration and e is the phase constant which 
orients the vibration in time. There is a difference, 
however, in the interpretation of a. In the plane wave 
case, a represents the maximum pressure change in 
the wave at all distances from the source; since a is a 
constant, all these pressure changes are equal. For 
spherical waves described by equation (42), however, 
it is clear that the maximum pressure change at the 
distance r is given by a/r, decreasing as r increases. 
The constant a is no longer the amplitude at all 
ranges, but merely the amplitude at the particular 
range r = 1. 


2.4 PROPERTIES OF SOUND WAVES 
2.4.1 Pressure versus Fluid \ elocity 

Plane Waves 

For a plane wave we have, from equation (4), 
da du 

dt dx’ 


where u is the particle velocity in the positive x 
direction. Because of equation (18), this equation 
can be transformed into 


dp du 

dt dx 


(43) 


From equations (21) and (18), there results the 
following expression for dp/dx: 


dp 

dx 



(44) 


Equations (43) and (44) will be used to derive the 
general relationship between the excess pressure and 
the particle velocity in a plane wave. Assume that 
as initial conditions the plane x = 0 has its excess 
pressure given by p(0,t) = p{t) between t = 0 and 
t = t 0 , and p(0,t) = 0 for all other values of t. Then 
the general solution of the plane wave equation, if 
we assume that the wave moves in the positive x 
direction, is given by 

p(x,t) = pit - 



By differentiating both sides of this equation with 
respect to t and also with respect to x, we obtain 



This means that 


dp 

dx 


l dp 
c dt 


(45) 


Combining equations (43) and (45) gives 



(46a) 


and by combining equations (44) and (45) 

7,(p - Pocu ) = 0. (46b) 

dt 

Equation (46a) means that (u — cp/«) is a func¬ 
tion of t alone; and equation (46b) implies that 
(p — pocu ) is a function only of x. But these two 
parentheses are proportional, differing by the factor 
( — p 0 c) since p 0 c 2 = k. Therefore, each parenthesis 
must be identically equal to some constant. This 
constant turns out to be zero for both since at any 
given point both p and u vanish before and after the 
disturbance has passed. The following relation be¬ 
tween particle velocity and pressure results: 

c 1 

u = - p = — p- 

K PqC 


(47a) 







20 


WAVE ACOUSTICS 


Equation (47a) can also be shown to hold good for 
the wave moving in the positive direction if the 
initial conditions are for the space interval 0 < x < x 0 
at the time t = 0. If the wave is moving in the nega¬ 
tive x direction, it can easily be shown by an argu¬ 
ment similar to the above that the pressure and par¬ 
ticle velocity are related by 



k Poc 

For the particular case of a plane harmonic wave, 
we have from equations (47a) and (47b) 


u 


± — cos 2wf(t — e). 

PoC 


The following interesting result stems directly from 
equations (47). If the particles in a plane wave are 
moving in the direction of wave propagation, they 
are in a region of positive excess pressure; if they are 
moving in a direction opposite to the route of the 
wave, they are in a region of negative excess pres¬ 
sure; and if the particles are not moving, they are in 
a region of zero excess pressure. Also, we can argue 
from equations (47) that if the initial conditions ful¬ 
fill neither 


nor 


u(x,0) = — p(x,0) 

PoC 

u{x,Qi) = ——p(x,0)> 

PoC 


then waves are propagated in both a positive and a 
negative direction from the initial source of pressure 
disturbance. 


Spherical Waves 

At great distances from the source a small section 
of a spherical wave approximates a plane wave. For 
this reason, many of the foregoing results can be re¬ 
written in a form valid for spherical disturbances far 
from the source. Since the mathematical proofs, 
though straightforward, are rather cumbersome they 
will not be reproduced here. 

For a general spherical wave far from the source, 
the following relation exists between the excess pres¬ 
sure and particle velocity: 

c 1 

u = ±-p = ± — p> 

K poC 

in analogy with equations (47). For a spherical har¬ 
monic wave it will be remembered that the maximum 
pressure change at the distance r from the source is 


given by a/r. Thus, for the case of a spherical har¬ 
monic wave, 

W = (48) 

p 0 c r 

The relations (47) are not necessarily true for the 
general solution of the wave equation (27). 


2.4.2 Acoustic Energy and Sound 
Intensity 

The vibration of the particles of a fluid disturbed 
by wave propagation is a process which involves both 
kinetic and potential energy. The energy of vibration 
of the sound source is propagated through the fluid 
along with the sound wave. In a perfect fluid where 
frictional heat losses are zero, the energy content of 
the wave is unchanged as the wave travels. The en¬ 
ergy passes from one region to another, “activating” 
the region through which the wave is passing. Thus, 
there are two quantities of interest. One is the energy 
found at any location as a function of time; the other 
is the rate at which energy is transported from one 
region to another as a function of time. In the follow¬ 
ing sections, both of these quantities are expressed in 
terms of the wave parameters we have introduced. 

The kinetic energy possessed by a volume element 
v, whose volume was w 0 at equilibrium, and whose 
speed is u, is given by 

Kinetic energy of v = ?p 0 v 0 u 2 . (49) 


The potential energy possessed by the volume ele¬ 
ment v is the work which was done on it to change its 
volume from Vo to v. This work can be calculated as 
follows: By equation (9), the relative change in 
volume produced by an infinitesimal alteration of 
condensation from a to a + da is just —da. The total 
volume change caused by this infinitesimal alteration 
of a is, to a first approximation, —v 0 da. The work 
done during this infinitesimal alteration is merely the 
pressure times the small volume change, that is, 
pv 0 da, which, because of equation (18), equals 
Kav 0 da. The total amount of work done on the volume 
element as its volume changes from v 0 to v can be 
obtained by integrating this infinitesimal amount of 
work between a condensation of zero and condensa¬ 
tion of a. 


Potential energy of 


v = v 0 kJ 


ada = %Voica 2 


Vo P~ 
2k 


(50) 



PROPERTIES OF SOUND WAVES 


21 


because of equation (18). By adding equations (49) 
and (50), we obtain 


rp , , , PoVoU- vo o 

1 otal energy of v = -- -f- — p~. 

2 2k 

By dividing equation (51) by the volume v 0 : 


(51) 


Energy density at ( x,y,z ) 


pom- V~ 
2 + 2k 


Po, 


= + K + < 


+ 


2k 


(52) 


We are now in a position to give a general expres¬ 
sion for the intensity of a progressive sound wave, the 
characteristic which determines its loudness. Inten¬ 
sity is defined for a general progressive wave as the 
amount of energy which crosses a unit area normal to 
the direction of propagation in unit time. Since the 
energ> r travels at the same rate as the sound pulse, 
the instantaneous rate of energy flow will be equal to 
the energy density at the point in question times the 
sound velocity at this point. The intensity will be the 
time average of the instantaneous rate of energy flow, 
or 


Intensity at ( x,y,z ) 


= Time average of c 


/p 0 u 2 p 2 \ 

\ 2 2 J 


Cpou 2 cp 2 

~~ 2 ~ + 27 


(53) 


where the bar over a quantity denotes the time 
average of that quantity. 

We shall now attempt to calculate the intensities 
explicitly for various types of sound waves. We shall 
first consider plane waves and spherical waves, and 
then more general waves. 


Plane Waves 

A plane progressive wave satisfies equation (47); 
and therefore equation (50) reduces, for that case, to 

Potential energy of v = ^poVoii 2 (54) 

which is exactly equal to the expression (49) for the 
kinetic energy of v. We therefore get the result that 
for a plane progressive wave the kinetic and potential 
energies possessed by any small volume element at 
any time are equal. The kinetic and potential energies 
attain their greatest values at the spots where the 
particle velocity and excess pressure have their max¬ 
ima or minima; and they vanish at the spots where 


the particle velocity and the excess pressure are both 
zero. 

Because of this equality of kinetic and potential 
energies for a progressive plane wave, equation (53) 
simplifies to 

Intensity = cp 0 u 2 = — (55) 

PoC 

by equations (47) and (26). 

In a plane progressive wave that is also harmonic 
the pressure is a sinusoidal function of the time with 
maximum value a. Since the average value of sin 2 6 
over a complete period is x /i, p- = a 2 /2. 

a 2 

Intensity = -- (56) 

2p 0 c 

Also, from equation (55) and the fact that u is also 
a sinusoidal function of the time, 

Intensity = §p 0 cu 2 max . (57) 

Spherical Waves 

For a spherical wave far from the source the 
formulas derived for plane waves are approximately 
true. We must be careful in applying them, however, 
to remember that the amplitude of the pressure 
vibration is no longer constant, but diminishes in¬ 
versely with distance. 

Using equations (57) and (48), we obtain 

Intensity = (58) 

ZpoC r 

for harmonic spherical waves where a is the maximum 
pressure change at a distance one unit from the 
source. 

Equation (58) is the familiar inverse square law of 
intensity loss for a spherical wave spreading out from 
a point source into an infinite homogeneous medium. 

Let F represent the amount of energy radiated by 
the source into a unit solid angle d in one second. 
Then 

Total rate of emission = AttF. (59) 


d Solid angle is the three-dimensional analogue to the 
ordinary, two-dimensional, plane angle. It measures the angu¬ 
lar spread of such three-dimensional objects as a cone, a light 
beam, or the beam of a radio transmitter. Its measure is de¬ 
fined as follows. Construct a sphere of arbitrary size with the 
apex of the solid angle as its center. The solid angle will then 
cut out a certain area of the surface of the sphere. This area, 
divided by the square of the radius of the sphere, is the meas¬ 
ure of the solid angle. It is dimensionless and does not depend 
on the sphere radius chosen. The unit solid angle is frequently 
called the steradian. The full solid angle, comprising all direc¬ 
tions pointing from the apex, has the value 4 t r. 





22 


WAVE ACOUSTICS 


We can calculate F by means of equation (58). 
Since a sphere of radius r has the area 4-n-r 2 , the total 
energy crossing such a spherical surface per unit of 
time is merely the intensity times this area: 

‘lira 2 

Rate at which energy crosses sphere --(60) 

PoC 

Because of the assumption of conservation of sound 
energy, equation (60) must be equal to the amount of 
energy radiated by the source per unit of time. 
Dividing equation (60) by 4x gives 


F = 


a 2 

2po c 


(61) 


General Sound Waves 


We now examine the transport of acoustic energy 
for the case of a general solution of the wave equa¬ 
tion (27). 

In the general case, it is useful to start with an 
equation of continuity for energy flow analogous to 
the exact equation of continuity (2) for mass flow. 
It will be recalled that equation (2) followed directly 
from the law of conservation of mass. The law of con¬ 
tinuity for energy flow will follow from the law of con¬ 
servation of energy in exactly the same fashion. For 
the mass density p, the energy density which may be 
denoted by Z is substituted. Also, for the instanta¬ 
neous flow of matter with components u x ,u v ,u z , the 
instantaneous flow of energy is substituted. The 
components of the instantaneous energy flow past 
normal unit area may be denoted by E x ,E y ,E z . The 
equation of the continuity for energy flow becomes, 
in analogy with equation (2), 


dZ T dE x dE u dE z 

dt L dx dy dz _ 


(62) 


Equation (62) is the mathematical expression of the 
assertion that the energy flow through a closed sur¬ 
face is equal to the decrease of energy inside this sur¬ 
face. A rather complicated argument must be used to 
calculate the components of energy flow E X) E y ,E z . 
Equations (21) are rewritten by using p = ko, as 



dp 

dx 


du,j dp 

' dt dy 



<>P 

dz 


(63) 


Also, from equations (4) and (18), we have the rela¬ 
tion 


1 dp 
k dt 


du x du y du z 

- 1 -- + - ' 

_ dx dy dz J 


(64) 


Multiplying the first equation of (63) by u x , the 
second by u y , the third by u z , and equation (64) by 
p, and adding them all up, we obtain 


Po 

L2 


(^X d" U 2 y A” U l) A” 


pr 

2k- 


d(pu x ) d(pu y ) d(pu z ) 

_ dx dy dz _ 


(65) 


Because of equation (52), we see that the left-hand 
sides of equations (65) and (62) are equal. Hence the 
right-hand sides are also equal, and we must have 


E x = pu x ; E y = pu y ) E z = pu z . (66) 


The instantaneous energy flow E is the resultant of 
its three components E x ,E y ,E z and is numerically 
equal to V 7 E\ + E\ + F\. Thus, we have the general 
result that 

E = pu. (67) 

According to equation (66), this energy flow is always 
along the direction of the particle velocity. 

The intensity I, which was defined as the time 
average of E, is therefore always given by the fol¬ 
lowing formula: 

/ - pu. (68) 


2.4.3 Complex Representation of 
Harmonic Vibrations 

The complex number e iw is defined by the equation 
e iw = cos w -f- i sin w 

where i 2 = —1. The one-dimensional harmonic vi¬ 
bration 

d = a cos 2wft 

can therefore be regarded as the real part of ae l2irft . 
Similarly, the vibration 

d = a cos 2irf(t — e) (69) 

can be rewritten as 

d = real part of ae l2nf{t ~ e) . 

The latter relation can be expressed in the following 
less cumbersome form 

D = ad 2irf{t ~ () (70) 

if the conventions are adopted that the actual phys¬ 
ical displacement is the real part of the complex dis- 














PROPERTIES OF SOUND WAVES 


23 


placement D and that the numerical value of this 
actual displacement is the real part of the right-hand 
side of equation (70). With this understanding, 
equations (70) and (09) represent one and the same 
physical process. 

The complex form for a vibration simplifies some 
types of calculations and will be frequently used in 
the remainder of this chapter. We notice that equa¬ 
tion (70) can be rewritten in the form 

D = Ae i2 ” ft (71) 

where .4 is the complex number ae~ riirft . A is called 
the complex amplitude of the vibration described by 
equation (69). It is apparent that 

A = a fcos 2irfe — i sin 2irft~\. 

Thus, the complex amplitude has a cos 2irfe as its real 
component and —a sin 2irfe as its imaginary com¬ 
ponent. 

As an example of the convenience afforded by the 
complex representation of a vibration, we shall use it 
to find the harmonic solution of the plane wave equa¬ 
tion (30). We assume tentatively that 

p(x,t ) = Ae 2H(ft + mx) (72) 

and see if we can find a value of m which will make 
equation (72) a solution of equation (30). Substi¬ 
tuting equation (72) into equation (30), we have 

(2irif)' 2 p = c 2 (2irim) 2 p. 

In other words, a value of m equal to f/c or —f/c 
makes the expression (72) a solution of equation (30). 
These two solutions are, explicitly, 

p = Ae 2HfD±(x/cn ; A = ae~ 2iri/ \ (72a) 

These two solutions, interpreted according to the 
convention of this section, are obviously identical 
with the “real” solutions (34). 

Similarly, a point harmonic source in an infinite 
homogeneous medium gives rise to spherical harmonic 
waves according to the equation 

p(r,t) = a = ae~ 2wih . (73) 

r 

2.4.4 Sound Sources 

The wave equation (27) governs the manner in 
which disturbances will be propagated in the interior 
of a fluid, but does not say anything about the initial 
disturbances themselves. In this section we shall con¬ 
sider the various types of initial disturbances which 
can be produced by sound sources. First we shall dis¬ 
cuss the quality of the sound put out by various 


sources, where quality refers to the frequency char¬ 
acteristics of the emitted sound. Next, since some 
sources radiate equally in all directions while others 
do not, we shall consider in a general way the 
directivity properties of sources. 

Frequency Characteristics 

Strictly speaking, the concept of frequency can be 
applied only to simple harmonic disturbances. A 
simple harmonic disturbance of the pressure in a fluid 
is described by an equation of the form 

p = a cos 2tt fit — e) 

and gives rise to what is called a pure tone. Most of 
the echo-ranging transducers used at present produce 
sounds which are very nearly pure tones, but cannot 
be heard by the ear because the frequencies are too 
high. 

If two or more pure tones are put into the water at 
the same time, the resultant is known as a compound 
tone. Some transducers, used mainly in research work, 
can produce compound tones. Any sound of this 
nature can be expressed as the sum of a finite number 
of harmonic vibrations. 

Many sources, however, produce in their immedi¬ 
ate vicinity an irregular change in pressure which 
cannot be represented as the sum of a finite number 
of sinusoidal vibrations. Such sound outputs are 
called noises. Ship sounds and torpedo sounds are 
examples of noises, and the reader can doubtless 
supply other examples. According to a mathematical 
theorem called the Fourier theorem, it is often possi¬ 
ble to represent such an irregular sound output as an 
infinite sum of simple tones, whose intensities, fre¬ 
quencies, and phase relationships are such that they 
add up to the given noise. If most of the component 
frequencies lie in a narrow frequency range, the sound 
is called a narrow-band noise; otherwise it is called 
a wide-band noise. 

Some types of echo-ranging gear put out a fre¬ 
quency-modulated signal. In this type of output, the 
pressure is at every instant a sinusoidal function of 
time, but the frequency changes during the signal in 
some designed way. In one type of frequency-modu¬ 
lated signal, called a “chirp” signal, the frequency 
increases linearly with time for the duration of the 
pulse: 

p = a cos 27 t[(/u + at)f\. 

In a typical chirp, 100 msec long, the frequency may 
increase from 23.5 kc at the beginning of the pulse to 
24.5 kc at the end of the pulse. 



24 


WAVE ACOUSTICS 


Directivity Characteristics 

So far we have been mainly concerned with the 
simple point source which gives rise to a spherically 
symmetric disturbance in the immediate vicinity of 
the source. It is called a point source because the re¬ 
sulting sound field is discontinuous at only one point 
of space, at the source itself. If the discontinuity is of 
a more complicated nature, as in the case of a line 
source, the sound field will not, in general, be spheri¬ 
cally symmetric in the neighborhood of the source; 
that is, the amounts of sound energy radiated into 
different directions will be different. In this subsec¬ 
tion, sources giving rise to sound fields that are not 
spherically symmetric are discussed. 

Double Sources. Suppose there are two point- 
sources, S 0 and So , one at ( Xo,yo,z 0 ) and the other at 
{xo ,yo,Zo), as indicated in Figure 6. The resulting 

P = P 0 + P 0 / 

k 

\ \ 



Figure 6. Resultant pressure produced by two sepa¬ 
rate sources. 

pressure at any one point P and time t will be the 
algebraic sum of the pressures that would be pro¬ 
duced by each source separately. That is, if / and f 
are the two frequencies emitted, e and t are the two 
phase constants, and A and A' are the two complex 
amplitudes, the resulting p(r,t ) is given by 

p = -e 27ri/c< — (r/c)] + 4e 2 ” /T '- (r7c)] . (74) 


Also, 

* A = ae- 2wift ; A' = aV 2 ” /v , (75) 

according to equation (71). 

We shall restrict our attention to the case of two 
sound sources situated on the x axis, one at the origin 
and the other a small distance s away. We assume 
that these two sources produce initial pressure dis¬ 
turbances of equal real amplitude and equal frequency 
and that the initial disturbances are opposite in phase. 


y 



Figure 7. Resultant pressure produced by double 
source 00'. 


This case, pictured in Figure 7, is a fairly good ap¬ 
proximation to many sources occurring in practice, 
such as a vibrating diaphragm. Because of these as¬ 
sumptions, the following relationships exist among 
the quantities in equations (74) and (75): 

a = a'; f = /'; e = 0; e'= (76) 

Also, A = a and A' = ae~ wl = —a because of equa¬ 
tion (75). 

Of particular interest is the extreme case where the 
distance between the two sources is very nearly zero, 
but where the real amplitude a of the individual dis¬ 
turbances is so large that the product as is an ap¬ 
preciable quantity. Such a combination of two single 
sources with very small separation and very large 
individual amplitudes is called a double source. A 
double source may be described by two quantities: 
the product as, and its axis, the direction of the line 
joining the two sources. 





PROPERTIES OF SOUND WAVES 


25 


With these assumptions, equation (74) becomes 


V 


= ®2«/[t-(r/c)] _ “2ri/Ct-fr'/c): 


(77a) 


where 

r 2 = x 2 + ?/ 2 + 2 2 ; r' 2 = (x 


s) 2 + 2/ 2 + z 2 . (77b) 


If F(r) is an arbitrary function of r, and if r and r' 
are very nearly equal, we have from simple calculus 


F{r) - F(r') 


, dF 
(r - r' — • 
dr 


The quantity r — r' may be calculated as a function 
of x,s,r as follows: 


r + r' 


2 r 


because r 


which equals sx/r from equation (77b). The quan¬ 
tity dF/ dr may also be calculated: 

dF dF dx dF dy dF dz 
dr dx dr dy dr dz dr 

As the origin changes from 0 to O' on the x axis, 
thereby changing r to r', the coordinates y and z of 
all points in space are unchanged. Thus, for all 
changes in r defined in this manner, 

dy = dz = 0 
dr dr 


so that 


dF 

dr 


dFdx 

dx~dr 


dFr 
dx x 


because of equation (36). Using these values of 
r — r' and dF /dr, 

F(r ) - F(r') = s— 
dx 

and equation (77) maj r be rewritten as 


p = ase 2nlft 


, d fl 
dxl_r 


-2*if(r/c) 


By calculating out the derivative of the bracket with 
respect to x, and by remembering that dr/dx = x/r, 
this equation becomes 

p = ase^'e- 2 ^-^ + - 

r 2 \ c r. 

If a is the angle between the x axis and the radius 
vector OP, x/r = cos a, and the preceding equation 
becomes 


p = ase 2 * ift e 


—2 r if(r/c) COS ^ l\ 

r \ c r) 


If r is very large compared with c/f, the second term 
in the brackets may be neglected, and as a result 

2 irfasi 2irfi[t—(r/c)l 

p = -cos ae 11 1 WJ . 


Replacing the factor as by b, we obtain the final 
result 


V 


cr 


(78) 


By comparing equation (78) with equation (73), we 
see that the pressure changes produced at great dis¬ 
tances by the double source are identical with the 
pressure changes produced by a single source, which 
is situated at the same place, vibrates with the same 
frequency, and has the following complex amplitude. 

2wfbi 

A = - L - cos a. (79) 

c 


It is clear from equation (75) that the real ampli¬ 
tude of the vibration is equal to the absolute value 
of the complex amplitude. From algebra, we know 
that the absolute value of a complex number A is 
just 's/aA, where A is the conjugate complex of A. 
Let a be the real amplitude corresponding to equa¬ 
tion (79). Then a is given by 


a — 


2t rfb 

- cos a. 


c 


(80) 


With this definition of a, the actual pressure dis¬ 
tribution defined by equation (78) is 

p(r,a,t) = a cos 2irf(t — • (81) 

Since p is harmonic, its square averaged over a com¬ 
plete period is one-half the square of its amplitude 
(80); from equation (58) we have for the intensity at 
the distance r and angle a: 



Thus, the sound intensity caused by a double sound 
source is directly proportional to the square of b and 
to the square of the cosine of the angle a of emission 
and is inversely proportional to the square of the 
distance from the sound source. 

Let F(a) denote the average rate at which energy 
is emitted in the direction a. It is clear from Figure 8 







26 


WAVE ACOUSTICS 


that this average emission per unit solid angle is 
given by 


F(a) 


I ( r,a ) r-dw 
du 


2ir-f' 2 b- 

Poc 3 


COS 2 a 


(83) 


where dco is an infinitesimal solid angle in the direc¬ 
tion a. The maximum value of F(a) occurs in the 
direction' of the x axis, for which 

F( ;0) = (84) 


Thus equation (83) can be rewritten as 

F(a) = F{ 0) cos 2 a (85) 



Figure 8. Rotation of wedge aa about axis. 


and equation (82) as 

F(0) cos 2 a 

a) = --— 

r- 


( 86 ) 


In order to find the total energy emitted by the 
double source in one second, we calculate the total 
energy traversing the surface of a sphere of radius r in 
one second. This is clearly equal to the rate at which 
the source is putting out power. To get this total 
energy, it is necessary to integrate the average energy 
flow (86) over the whole sphere. Such an integral is 
in general multiple, but in this particular case it can 
be expressed as a single integral because the energy 
flow depends only on a. First consider the average 
rate at which energy is flowing through the small 
area element intercepted on the sphere by the two 
cones defined by the angles a and a + da, as in 
Figure 8. This small element of the sphere has the 
area 2nr 2 sin ada ; and, therefore, it intercepts a solid 
angle of 2ir(sin a)da units. By equation (86), the 
average rate of energy flow through this element is 

F( 0) COS 2 a ■ 2ir sin a • da. 


The total emission in one second is this average rate 


of energy flow integrated between the angles 0 and tt; 
that is, 

/»jt/2 

Rate of emission = 2 I F( 0) cos 2 a • 2ir sin a • da 

Jo 

= (87) 

3 

It will be remembered that F(0) is the maximum rate 
of emission per unit solid angle, by the double source. 

All sound projectors have pattern functions which 
describe the distribution of sound energy emit¬ 
ted in different directions. A general direction in 
space can be defined by the two coordinates (0,0), 
where 0 is the angle of elevation of the direction OP 
relative to the horizontal xy plane, and 0 is the polar 
angle in the xy plane between the x axis and the 
projection OP' , as in Figure 9. Let F(0,0) be the 

y 



Figure 9. Coordinates specifying direction OP. 


emission per unit solid angle in the direction OP, and 
let F max be the emission per unit solid angle in the 
direction of maximum emission, called the acoustical 
axis of the projector. Then the pattern function 
6(0,0) is defined by 

F(0,0) = F max 6(0,0). 

If we take the acoustical axis in the direction (0,0), 
this becomes 

F(0,0) = F(0,0)6(0,0). (88) 

The pattern function 6 clearly depends only on the 
nature of the projector. 












SOUND WAVES IN A VISCOUS FLUID 


27 


In analogy with the result (86) for the simple case 
of axial symmetry, 


l(r,0,<t>) 


F( 0,0) b(9,<t>) 
r- 


(89) 


at a distance r from the source much greater than a 
wavelength. 

The rate at which the projector emits energy in all 
directions must be exhibited as a double integral in 
this general case. The area element on the sphere of 
radius r, intercepted between ( d,4> ) and (0 + dO, 
4> d<f>), is r- cos Oddd<t> ; and the solid angle inter¬ 

cepted by this area element is cos dd6d<t>. Thus, in 
view of equation (88), 

Emission through area element 

= F(0,0) b{6,4>) cos 6ddd<f> (90) 

and therefore 

Total rate of emission 

= E(0,0) | d<f> ( b(d,<t)) cos Odd. (91) 

Equation (91) can be put in the following form* 
which is directly comparable to the law of emission 
(59) of a point source: 

Rate of emission = 4 ttE(0,0)5 (92) 

where 

S = — I d<t> I b(d,cf>) cos Odd. (93) 

-lird-V d -7T/2 

The factor 5 is a constant depending on the nature of 
the source and may be called the directivity factor of 
the source. For the point source of equation (59), this 
directivity factor is 1; while for the double source of 
equation (87) it is 


In the derivation of the wave equation (27) for a 
perfect fluid, we used the equation of continuity (4), 
the equations of motion (15), the equation of state 
(18), and the law of forces (16). The equation of 
continuity, the equations, of motion, and the equation 
of state, it will be recalled, apply to any fluid, whether 
it is perfect or viscous. The law of forces (16), how¬ 
ever, is valid only for a perfect fluid. The exact law 
of forces operating in a viscous fluid is quite difficult 
to derive since it depends on the theory of viscous 
fluid flow. It is sufficient to say that this complicated 
law of forces, combined with equations (4), (18), and 
(15), can be used to derive a general wave equation 
for viscous fluids, analogous to equation (27). Under 
the assumption that the resulting pressure distribu¬ 
tion in the viscous medium depends only on the co¬ 
ordinate x, this general equation reduces to 3 

+ (94) 

dt 2 po dx 2 3po dxdt 


where p is the coefficient of shear viscosity. Equation 
(94) is the plane wave equation for a viscous fluid. In 
the absence of viscosity (p = 0), equation (94) re¬ 
duces to equation (30). 

Let us see whether we can find a solution to equa¬ 
tion (94) of the form 


p = Ae 2*H/t+mx)' 

By substituting this expression for p into equation 
(94), we obtain 


m 2 


r- 



4m 

3p 0 


2mf 


(95) 


If c and a are defined by 


2.5 SOUND WAVES IN A VISCOUS FLUID 

In a homogeneous perfect fluid, the decrease of 
sound intensity with increasing distance from the 
source is due only to spreading according to the 
inverse square law (58). However, sound intensity 
measurements show clearly that the intensity loss in 
the ocean tends to be much greater than the value pre¬ 
dicted by equation (58). These extra losses above the 
theoretical loss, (58), due to the fact that the ocean 
is not a homogeneous perfect fluid, are called trans¬ 
mission anomalies or, more loosely, attenuations. In 
this section, we shall derive some results for sound 
intensity in a viscous fluid and see how much of the 
observed attenuation can be ascribed to fluid 
viscosity. 


c 2 


a 


K 

Po 


8 2 m/ 2 
_7r —: 
3 poc 3 


the relation (95) becomes 


m 


c 


1 + 




(96) 

(97) 



, (98) 


according to the binomial theorem, if we assume that 
a is so small that a 2 and higher terms can be neglected. 

In order to get the case of waves propagated in the 
positive x direction, the negative sign of equation 
(98) must be chosen and 2 iritnx becomes 


2mmx = 


9 •/ 

— 2m ~x — ax. 


c 







28 


WAVE ACOUSTICS 


The corresponding solution of equation (94) is there¬ 
fore 

p = Ae Zirift e ~ 2nif(x/c) e~ ax . 

We can write this expression for p in the form 

p = Ae~ ax e 2irim - (x/cn . (99) 

For n = 0, a vanishes, and equation (99) reduces 
to 

p = Ae »'<«-(*/«)] ( 100 ) 

which is just the solution for plane waves propagated 
harmonically into a perfect fluid. By comparing 
equations (99) and (100), we see that the effect of 
viscosity is to cause the amplitude of the pressure 
vibration to decay exponentially with distance, by 
the factor e~ ax , where a is the positive real number 
defined by equation (97). A vibration of the type 
of (99) is referred to as a damped vibration, and e~ ax 
is called the damping factor. 

To see whether this energy loss due to shear 
viscosity is the cause of the attenuation observed in 
the sea, one can first calculate a for sea water, by 
using the known values of p n ,c,p for sea water, and 
the known frequency/ of the sound source. The in¬ 
tensity loss is measured between two points so far 
from the sound source that the wave propagation be¬ 
tween those two points approximates plane wave 
propagation. Then this observed intensity loss is 
compared with the theoretical intensity loss calcu¬ 
lated from equations (97) and (99). It is found that 
only for very great frequencies (much higher than 
100 kc) can an appreciable fraction of the observed 
attenuation be ascribed to shear viscosity; at lower 
frequencies, the theoretical loss from viscosity makes 
up only a very small part of the observed attenua¬ 
tion. Thus other causes must be sought for the extinc¬ 
tion of sound energy in the sea. The sound transmis- 
tion studies of Section 6.1 of NDRC have had as one 
of their primary objectives the discovery of the 
factors governing the intensity loss of sound in the 
sea. Although some progress has been made, the 
problems of attenuation in the sea have by no means 
been completely solved (see Chapters 5 to 10). 

Since the observed attenuation of sound in the sea 
is much greater than the value indicated by equation 
(99), it appears possible that there is another type of 
viscosity, in addition to the classical shear viscosity, 
which may be responsible for part or all of the re¬ 
maining attenuation. The classical theory of the 
flow of viscous fluids is based on Stokes’ hypothesis 
that frictional forces within a fluid arise only from a 


change in the shape of a volume element; in other 
words, that a change in the size of a volume element, 
if its shape remained unaltered, would meet no re¬ 
sistance. The concept of a compression viscosity has 
been suggested to represent the resistance of the 
fluid to pure volume dilatation. Such a compression 
viscosity would not be discovered in a stationary flow 
of the type employed to measure shear viscosity be¬ 
cause in these experiments the fluid acts essentially 
as an incompressible fluid. But in the transmission 
of sound this conjectural compression viscosity would 
contribute a term to the expression for a which would 
also be proportional to the square of the frequency/. 

Actual determinations of the constant a at many 
different frequencies show that between 0 and 100 kc 
the attenuation increases less rapidly than the 
square of the frequency. There are no theoretical 
grounds for assuming any power law for the depend¬ 
ence of attenuation on frequency. If a power law is as¬ 
sumed, the empirical curve is best fitted by a 1.4th 
power dependence, but even this best fit is poor. It 
thus appears that factors other than viscosity must 
account for much of the attenuation of sound ob¬ 
served in the sea. 


2.6 EFFECT OF A BOUNDARY 

2 . 6.1 Conditions of Transition and 
Boundary Conditions 

We shall now return to the assumption of a perfect 
fluid and turn our attention to the effects of bounda¬ 
ries. Consequently, we now drop the assumption that 
waves are propagated in a single homogeneous infi¬ 
nite medium. Instead, we shall consider the case that 
all space is filled up by two different homogeneous 
media separated by a plane, which we choose as the 
plane y = 0. For the one medium (the sea), at y < 0, 
we denote the density, excess pressure, bulk modulus, 
and sound velocity by p, p, k, and c respectively; for 
the other medium, air, for example, at y > 0, we call 
these quantities p u p u ki, and ci. 

It is necessary, from a physical point of view, to 
assume that the pressure in both media is the same 
at the boundary. Otherwise, the force per unit mass 
at the interface would become infinite. We have, 
thus, 

p = pi at y = 0. (101) 

Also, if the two media are to remain in contact with 
each other at all times, the displacements normal to 



EFFECT OF A BOUNDARY 


29 


the boundary must have the same value in both 
media at the boundary. In symbols, if (S x ,S y ,S t ) are 
the components of particle displacement in the first 
medium, and (Si x ,Si v ,Si z ) are the components of 
particle displacement in the second medium, 


S y = Siy at y = 0. (102) 


No restrictions of the form of (102) can be placed on 
the displacements S x and S z because displacements 
parallel to the boundary will not cause loss of con¬ 
tact. 

Since equation (102) holds for all time, the time 
derivatives of S v and Si v must also be equal at the 
boundary; in other words, 


Uy ^ Uly\ 


6Uy 

dt 


dUly 

dt 


y = o. 


(103) 


Because of equation (17), equation (103) implies 


or 


1 dp 
P dy 
dp = 
dy 


I d pi 
Pi dy 


aty = 0 


P dpi 

- — at y = 0. 
Pi dy 


(104) 


We shall call equations (101) and (104) conditions 
of transition. In the general case, the propagation in 
one medium depends on the exact nature of the 
propagation in the other medium, because of the con¬ 
ditions of transition. In the case of the sea, however, 
conditions are often such that we can ignore the exact 
propagation in the surrounding medium; the transi¬ 
tion conditions of the type (104) may then be re¬ 
duced to boundary conditions for the sea itself. In the 
next section, the conclusion is reached that at the 
yielding boundary between sea and air the follow¬ 
ing condition holds: 

p = 0 (105) 


and that at the solid boundary between sea and rock 
bottom we always have, approximately, 


— = 0. (106) 
dy 

Relations of the type of (105) and (106) will, in many 
cases, suffice for calculating the sound field in the 
medium of interest. By use of such boundary condi¬ 
tions explicit consideration of the sound field beyond 
the boundaries may be made unnecessary. 


2.6.2 Reflection and Refraction 
of Plane Waves 

Consider now what happens to a plane wave when 
it hits the plane boundary y = 0 between two dis¬ 


similar media, in one of which the sound velocity is c, 
and in the other of which it is ci. For generality, we 
assume that the direction of propagation of the inci¬ 
dent wave is oblique to the boundary, making an 
angle 0, with the normal to the plane boundary. We 
can also assume, without losing generality, that the 
direction of propagation is parallel to the xy plane; 
y represents the vertical direction positive upward, x 
a horizontal direction, and z a front-back direction, as 
in Figure 10. 



Figure 10. Splitting of plane wave at boundary be¬ 
tween two media. 

Since the incident wave is plane, it may be de¬ 
scribed by the equation (72a) with x replaced by 

x sin di + y cos di, 

in view of the oblique direction of propagation. 
That is, for the incident wave, 

2vif ( t _ xBin0 i +yco B e a 

Pi = Ai e c , (107) 

where pi represents the sound pressure of the inci¬ 
dent wave, and A, its complex amplitude. 

We can consider that the incident wave terminates 
its existence when it hits the boundary and expends 
its energy in producing a disturbance of the interface. 
Thus the boundary will act as a sound source, which 
vibrates with the frequency / of the incident wave. 
The vibration of the interface will send out sound 
waves of the frequency / into both media. We shall 
assume that these two waves are plane waves; this 






30 


WAVE ACOUSTICS 


result is intuitively apparent, but can be proved only 
by a long tedious argument. 

For brevity, the wave propagated by the boundary 
into the second medium will be called the transmitted 
wave , and the wave propagated back into the first 
medium is called the reflected wave. We shall now cal¬ 
culate the amplitudes and directions of propagation 
of the transmitted and reflected waves? 

The pressure and complex amplitude of the trans¬ 
mitted wave are denoted by p, and A the same 
quantities for the reflected wave are denoted by 
p r and A r . Let the transmitted wave have the direc¬ 
tion 0,, relative to the normal, and the reflected wave 
have the direction 9 r , as indicated in Figure 10. The 
angles 6, and 8 r are usually called the angle of refrac¬ 
tion and angle of reflection, respectively, and the 
angle 0,- is called the angle of incidence. 

Because the reflected and transmitted waves are 
plane, 

o-ri/ff _ — -rsinfl,— yeos8 r \ 

p r = A r e ' c , (108) 

2 w {f / f _ Jsin9t+j/cos9i j 

Pi = A t e C1 • (109) 

The sign of y is different in equations (108) and (109) 
because in equation (108) y decreases with the time 
on the wave front; in equation (109) it increases. 

Equation (109) gives the resultant total pressure 
in the second medium. The resultant pressure in the 
first medium is the sum of the pressures of the inci¬ 
dent and reflected waves, which is obtained by adding 
equations (107) and (108). Denoting the resultant 
pressure in the first medium by p, we obtain 

P = Pi + Pr = A { e c 

2 ri}(i- x s-nflr -1/CO6 0, ) (HO) 

+ A r e c 

The pressure must be the same on both sides of the 
boundary. Therefore, p, + Pr = Pt at y = 0; that is, 

A ie K c + Are K c = A,e~ n Cl 


Tift 


.4 ,e 


-2 Kif- 


x + A r e 


x — A t e 


-2 «/- 


= 0 ( 111 ; 


for all values of t and x. Therefore, the bracket itself 
must be zero. Furthermore, the sum of three har- 


( 112 ) 


monic functions of x can vanish for all values of x 
only if their periods are the same. It follows that 
sin 8, sin 0,- sin 8 r 

Ci c c 

The second equation of (112) implies that 0, = 8 r ; 
that is, the angle of incidence is equal to the angle of 
reflection. The first equation may be rewritten as 

sin 0, c . 

- 7 - 7 =-’ (113) 

sin 8 t Ci 

a relation which is well known in optics as Snell’s law. 

Because of equation (112), the exponential factor 
is the same / for all three terms in the bracket of 
equation (111) and can be divided out, giving 

A t = Ai + A r . (114) 

The individual amplitudes .4, and A r are calculated 
by making use of the transition conditions (104). By 
calculating dp/dy from equation (110), and dp t /dy 
from equation (109) and by substituting these values 
into equation (104), we obtain 

*4, COS 0; 2 *if(t- x JB°±) Ar COS 0 r 2 

-£ C - -g c 


p A t COS 81 2t if (t- 

- e 

Pi Ci 


(115) 


In view of equation (112), the exponential factor is 
the same for all three terms and may be divided out. 
Also, 0, = 0 r . Thus, equation (115) becomes 


cos 0 P cos 0, 

- (Ai - Ar) = -At - 

C Pi Cl 


:ii6) 


Equations (114) and (116) are two linear equations 
in A t and A r . Bv solving them in terms of Ai, the 
amplitude of the incident wave, and by replacing 
Ci/c in the result with its equivalent from equation 
(113), 

PiCi cos 8i — pc cos 0 ( 


Ar = Ai 

At = Ai 


PiCi cos 8 i + pc cos 81 
2piCi cos 8i 


(117) 


(118) 


PiCi COS 0; + pc COS 81 

To eliminate the angle 0, from equation (117), equa¬ 
tion (113) is used which can be transformed by 
trigonometric identities into 


COS 0 ( 1 / Ci\ 

- = 1/ 1 + tan 2 0,1 1- -) = B. 

cos 8i r V c 2 / 


Thus, equation (117) becomes 

4r _ PlCl — pcB 

At piC\ + pcB 


(119) 




















EFFECT OF A BOUNDARY 


31 


Equation (119) gives interesting results when it is 
applied to the case of a sound wave in water hitting 
the surface separating water from air. The numerical 
values are (subscripts for air; no subscripts for 
water): 



Cl Pl 


By substituting these values into equation (119), 

Ar _ 1 - 3,31 lVl + 0.95 tan 2 ^ 

A i 1 + 3,31 lVl + 0.95 tan 2 0 t 

For perpendicular incidence 0, vanishes, and A r /Ai 
differs from —1 by less than one part in a thousand; 
for greater values of 0, the approximation to —1 is 
even better. A wave in water reflected by air thus 
preserves its real amplitude almost exactly, that is, 
almost all the energy in the incident wave remains in 
the water. But it reverses its phase; this means that 
p T - —pi at the boundary. This conclusion, that the 
resulting total pressure at this type of interface 
should be very nearly zero, was called a boundary 
condition in Section 2.6.1. The derivation of this 
section furnishes the justification for assuming this 
boundary condition, which was stated without rigor¬ 
ous proof in Section 2.6.1. Equation (119) provides 
an estimate of the error caused by replacing transi¬ 
tion conditions at a boundary with the more simple 
boundary conditions. In the case of the interface 
separating water and air this error is clearly very 
slight. 

Another interesting case is the incidence of under¬ 
water sound on a hard bottom like solid rock or 
tightly packed coarse sand. The treatment of sound 
waves in solids is rather more involved than the 
treatment of sound waves in fluids because a solid 
has two different kinds of elastic forces: those which 
resist changes in volume; and those which resist 
changes of shape (bulk modulus and shear modulus). 
Consequently, two different kinds of propagation of 
sound are possible in a solid. The two types are 
usually referred to as longitudinal waves and trans¬ 
verse waves. In the oblique incidence of underwater 
sound in a water-solid interface both types of waves 
are generated in the solid and the transition condi¬ 
tions are, therefore, more involved than those dis¬ 
cussed previously. If, however, the solid is quite 
rigid — that is, if both bulk modulus and shear 
modulus are appreciably greater than the bulk 
modulus of water — then it may be assumed, in good 
approximation, that the interface will not permit 


displacements perpendicular to itself. In other words, 
u v will vanish approximately. If u v at the interface is 
zero, then its time derivative vanishes as well; and 
by reason of the equations of motion (17), 
dp 

— = 0 at y = 0. 

dy 

This is the boundary condition which is often as¬ 
sumed in the treatment of reflection from a hard 
bottom. If this boundary condition is realized, it can 
be shown that the incident and reflected waves will 
have equal amplitude and the same phase. Thus, 
when sound is reflected from a rock bottom, almost 
all the energy of the incident wave will be found in 
the reflected wave. For a soft bottom like mud, this 
boundary condition will no longer be satisfied, even 
in approximation, and considerable sound energy may 
be lost by transmission through the interface. 

2.6.3 Homogeneous Medium with 
Single Boundary 

Point Source Near Sea Surface 

We shall now solve the problem of finding the solu¬ 
tion of the wave equation which satisfies the bound¬ 
ary condition p = 0 at the interface y = 0, and cor¬ 
responds to a sound wave radiated by a point source 
at the depth h. This situation is illustrated in 
Figure 11. The depth of the ocean is assumed to be 


y 



Figure 11. Pressure produced at location P by sound 
source 0. 


infinite. The initial conditions are specified by the 
assumption that in the immediate vicinity of the 
source, that is, for points whose distance from the 
source r, 

r = x 2 + z- + (y + h) 2 

is very small, the pressure satisfies the relationship 
rp(r,t ) = F(t). (120) 










32 


WAVE ACOUSTICS 


If it were not for the boundary condition p = 0 at 
y = 0, the problem would be solved by means of the 
expression 

p{r,t) = (121) 

We shall have to modify this solution in order to 
satisfy the boundary condition as well. To this end, 
we resort to a trick. We solve a fictitious problem, 
one in which a source exactly like the first one is lo¬ 
cated at a distance h on the other side of the inter¬ 
face with the water extending through space and 
with the initial conditions 

r'p'(r',t) = -F(t) ( 122 ) 

at points very close to the new source. This problem 
has the solution 

P'(r',t) = ^ F (t - ( 123 ) 

where 

r' = y/x 1 + z 2 + (y — h) 2 . 

Clearly, since r = r' at y = 0, we have p + p' = 0 
at x = 0. Thus, the wave given by the sum of the 
two disturbances described by equations (121) and 
( 123 ), or 

Fit ~ (r/c)] _ Fit - (r'/c)] 
r r' 


satisfies the imposed boundary conditions. Also, 
equation (124) satisfies the initial conditions (120) 
because the expression (124) can be rewritten as 


rp _ - 0 - - 9 


which reduces to equation (120) in the vicinity of the 
actual source, where r « 0. Finally, equation (124) 
satisfies the wave equation itself since the difference 
of two solutions of that equation is itself a solution. 

If the source S executes a harmonic vibration, the 
solution (124) becomes 


V 


= a 


cos 2wf[t — (r/c)] 
r 


cos 2 ivf[t - (r'/c)]] 
r' r 


(125) 

Formula (125) fully describes the effect of surface re¬ 
flection on harmonic waves emitted by a single source 
under the assumptions that air has negligible density 
and elasticity and that the sea surface is a perfect 
plane. 

From the method of construction of the solution 
(124), it is possible to deduce that there should be a 


zone of low intensity near the surface. The reason is 
that, at points near the surface, r and r' will be nearly 
identical; and the two resulting fictional pressures 
will almost balance each other. This type of destruc¬ 
tive interference near the surface is called the Lloyd 
mirror effect or image interference effect. The next few 
paragraphs will discuss the width of this low intensity 
zone and the intensity within this zone. 

Consider the intensity measured by a receiver at 
the depth hi, located at a horizontal range R from 
the source, as in Figure 12. We also assume that R 



Figure 12. Fictitious scheme for solving wave equa¬ 
tion and surface boundary conditions. 


is so large compared with h and hi (the depths of 
source and receiver) that second-order products of 
h/R and hi/R may be neglected. 

By applying the Pythagorean theorem to Figure 
12, then 


, (hi - h) 2 


= flj/l + 


(hi + h) 2 
R 2 


Since h/R and hi/R are small, these equations may 
be rewritten as 


r = 7?|l + 


If (hi - h)* lj 

2L R 2 Jf 


-4 4^1 

and as a result 


(126) 



l _ l ( hl + h \ ( 127 ) 

l - 2 \ R ) 

r' ~ R 


because 1/(1 — e) = 1 + e if e is small. Putting 
these in equation (125), we obtain 
























EFFECT OF A BOUNDARY 


33 


plus negligible terms, which may be rewritten as 
r + 


a / 

p = fii- 2sm 


2tt/ t 


2c 


2nf 


CK)] 


-hi). 

by the trigonometric identity for the difference of two 
cosines. This equation reduces approximately, be¬ 
cause of equation (126), to 

r - -I sin WOf ] sin M -!)] • (i28) 


At the point P, equation (128) tells us that the 
amplitude of the pressure variation with time is 

, 2 a . hih 

Amplitude = — sin 2 ir — (129) 

R R\ 

where X = c/f is the wavelength of the sound. The 
resulting sound intensity at P, which is proportional 
to the square of the maximum acoustic pressure, will 
be very small if the argument of the sine in equation 
(129) is small. That is, the intensity will be low if 
hih/\ is very small compared with R, or, in other 
words, if 

hi<<— • (130) 

h 


Assuming that equation (130) holds, the sine in 
equation (129) will be approximately equal to its 
argument, and we have 


Amplitude = 


■iirahhi 

R-\ 


In terms of the intensity, this means 

lGir 2 a 2 /i 2 /ii / 1 \ 

Intensity cc -—- )■ (131) 

That is, in the layer of poor sound reception the sound 
intensity falls off inversely as the fourth power of the 
horizontal range at great ranges. 

For smaller values of horizontal range R, we find 
that the amplitude vanishes wherever the argument 
of the sine in equation (129) is an integral multiple of 
7r, or, in other words, where 


2hh\ 

IK 


= j,j = 0,1,2, 3,- 


while the amplitude will show greatest values in the 
neighborhood of those points where the argument is 
7 t/2, 37t/ 2- • •; in other words, where 


4 hhi 

IK 


1, 3, 5,• • • 


This sequence of interference minima and maxima is 
called the image interference pattern. 

The image interference effect described here is only 
occasionally observed in the sea for reasons which are 
discussed in Section 5.2.1. 


Point Source Far from Sea Surface 


We assume now that the source is located so far 
from the surface that the sound waves near the sur¬ 
face can be regarded as plane waves. Only incident 
waves propagated purely in the y direction are con¬ 
sidered, that is, normal to the surface. Then equa¬ 
tion (125) has to be replaced by 



— cos 2 irf 



■ (132) 


By applying to equation (132) the trigonometric 
formula for the difference of two cosines, we obtain 

p = —2a sin 2ir~ sin 2irft. (133) 

X 


We notice a very curious thing about the disturb¬ 
ance described by equation (133). The acoustic pres¬ 
sure is zero over the entire fluid when ft is any in¬ 
tegral multiple of }■%. Further, the acoustic pressure 
is zero for all time at points where y/\ is an integral 
multiple of Thus we see that the interference be¬ 
tween two plane waves of equal amplitude and of the 
same frequency traveling in opposite directions pro¬ 
duces, at least in this case, a disturbance of the 
medium for which at any instant all points have 
identical, or opposite phase. We no longer have pro¬ 
gressive waves, but a phenomenon which we call 
stationary or standing waves. The points where the 
amplitude is zero for all time are called nodes ; the 
points where the amplitude term of equation (133) 
is a maximum are called loops or antinodes. 

This state of affairs is permanent as long as the 
source keeps vibrating. The nodes are permanent re¬ 
gions of silence; and the loops are permanent regions 
of maximum pressure amplitude. Such a state, in 
which all points of the medium perform vibrations 
of the form sin 2irft with an amplitude dependent on 
position is called a stationary state of the medium. 


Reflection from Sea Bottom 

If water is separated by the plane y = 0 from a 
medium with a density much greater than its own, 
the boundary condition which must be fulfilled at 
this plane is 













34 


WAVE ACOUSTICS 


It turns out that a solution synthesized as was equa¬ 
tion (124), but with a plus sign in equation (122) in¬ 
stead of a minus sign, will satisfy this boundary con¬ 
dition. This solution is 


V 


F\ t - 


+ 


{' ~ 1 


(134) 


We verify that equation (134) satisfies dp/dy = 0 by 
differentiating equation (134) with respect to y, and 
noting that dr/dy = —dr'/dy at y = 0. 

We now examine the possibility of stationary 
states for the case where the boundary is a hard sea 
bottom. Again, we assume a harmonic source so far 
from the bottom that waves reaching the bottom are 
plane and we assume perpendicular incidence. Then 
equation (134) must be replaced by 


p = a 


COS 2 irf 


H) 


+ cos 2 tt/I t + 


(135) 


which, by trigonometry, reduces to 


p = 2 a cos 27r- sin 2irft. (136) 

A 

We easily see that equation (136) also represents a 
stationary state of our fluid. The nodes of utter 
silence are situated where cos 2w(y/\) disappears, 
that is, at y = A/4, 3X/4, 5X/4,•••; the loops 
of maximum sound intensity are located where 
cos 2iv(y/\) equals +1, that is, at y = 0, X 2, X, - • 


2.7 NORMAL MODE THEORY 

2.7.1 Plane Waves in a Medium 
with Parallel Plane Boundaries 

The problem of sound propagation in a medium 
bounded on two sides is extremely complicated and 
cannot be solved in general. The difficulty lies in the 
fact that the solution must satisfy not only the wave 
equation, but also the initial conditions and the 
boundary conditions at each boundary. 

In Section 2.6 it was shown that certain definite 
and instructive results could be obtained for the case 
of a single boundary by considering the case of plane 
waves and assuming (1) perpendicular incidence and 
(2) an infinite change in density at the boundary. 
The result was a standing wave pattern whose geo¬ 
metrical properties depended on the wavelength and 
whose maximum amplitude depended on the energy 
in the incident wave. 


We shall keep these two assumptions in this sec¬ 
tion and shall first find out under what conditions a 
stationary wave pattern of any sort can be set up in 
our bounded medium. The general expression for a 
standing wave pattern is 

p = >Ky) cos 2irf(t — e) (137) 

where \p is any function of y. In other words, equa¬ 
tion (137) means that all points of the fluid perform 
vibrations with the same frequency / and phase con¬ 
stant e, but with amplitude \p(y) depending on the 
position coordinate y. The immediate problem is to 
find out what sort of functions i p(y) are necessary to 
make equation (137) a solution of the plane wave 
equation 


(138) 


d 2 p = c ,^p 

dt 2 dy- 

and also a solution of the boundary conditions on 
the boundaries y = 0 and y = L. These boundary 
conditions are either equation (139a), (139b), or 
(139c). 

p = 0 at both y = 0 and y = L 


dp 


p = 0 at y = 0; — = 0 at y = 
dy 

— = 0 at both y = 0 and y 

dy 


= L 


(139a) 

(139b) 

(139c) 


It is immediately apparent that the condition for 
equation (137) to satisfy the plane wave equation 
(138) is 


(P\l/ 4ir~ 

— -|-* = 0. 

dy 2 A 2v 


(140) 


First consider the case of the boundary conditions 
(139a). The boundary conditions (139a) can be re¬ 
stated as 

*(0) = 0, \p{L) = 0. (141) 

The problem is thus reduced to the case of finding the 
solution of an ordinary differential equation with 
boundary conditions on both ends of an interval 
0 ^ y ^ L. 

Equation (140) is a simple differential equation 
whose general solution is well known to be 


• 27T 2tt 

> Ky) = A sin —y + B cos — y 
A A 

where A and B are arbitrary constants. 
The condition \p( 0) = 0 implies that B 

2tt 


fiy) 


A sin — y. 
X 


(142) 


0 and 


(143) 







NORMAL MODE THEORY 


35 


Since equation (143) must satisfy \J/{L) = 0, 


L 

sin 27r- 
X 



1 2 3 
2 ’ 2 ’ 2 ’ 


(144) 


This means that under the given boundary condi¬ 
tions there cannot be a stationary state of the type 
of (137) unless the wavelength X has one of a number 
of definite ratios to the depth L. The ratio X/L must 
be either 2/1, 2/2, 2/3, 2/4, or in general 2/j. A set 
of wavelengths X; can be defined by 


Xy — .L, j — 1, 2, 3, • • • (145) 

Then, if the actual wavelength in the problem is 
equal to one of these Xy, the expression (143) will 
satisfy equation (141); and the stationary state equa¬ 
tion (137), with this value of \p, will satisfy both the 
wave equation (138) and the boundary conditions 
(139a). If the actual wavelength is not equal to one 
of the Xy, then there can be no stationary state in the 
given medium in which the wave planes are parallel 
to the interfaces. 

Mathematically, all this means that the total 
problem defined by equations (138) and (139a) can 
be solved only if the coefficient of \p in equation 
(140) has certain definite values a, defined by 


ay = 


47T 2 

v’ 


(146) 


These values, ay, are called characteristic values. The 
solutions (143) corresponding to them, namely 


2^r 

\pj(y) = A sin —y = A sin Va, y (147) 
A 

are called characteristic functions of the problem. 
Clearly, every characteristic value ay corresponds to 
a possible frequency /y and wavelength Xy ; by pos¬ 
sible is meant that it gives rise to a stationary state, 
or normal mode of vibration. The characteristic func¬ 
tion \pj determines the distribution of acoustic pres¬ 
sure within this normal mode of vibration. 

If the boundary conditions are changed, the char¬ 
acteristic values and characteristic functions change 
also, although the differential equation which \p must 
satisfy remains equation (140). If the boundary con¬ 
ditions (139b), which correspond most closely to 
actual conditions in the sea, are assumed, it is found 
that, by methods similar to those described pre¬ 
viously, a normal mode can arise only if 


L 13 5 
X - 4’ 4 ’ 4’" ‘ 


(148) 


That is, 

4L , 

Xy = — where j = 1, 3, 5, - • • (149) 

J 

We notice that the characteristic wavelengths Xyare 
different from the first case. The characteristic values 
and functions can be calculated by using equations 
(146) and (147) and by remembering that the Xyinust 
now be taken from equation (149). 

Clearly a sum of two normal modes also satisfies 
the differential equation (138) and the imposed 
boundary conditions. 

Suppose we have the general case where the 
boundary conditions determine an infinite number 
of normal modes 

P = c xpj (y) cos 2wfft - €y), j = 1, 2, 3, • • •. (150) 

Suppose we have initial conditions on y at t = 0, 
of the nature 

p(y, 0) = DUv), (i5i) 

where D is some constant. For equation (150) 
to satisfy the initial conditions (151), we must 
choose / = k, e. = (l/27r/„) arc cos D/c. Since this can 
always be done, we have the result that if the initial 
pressure disturbance is a multiple of one of the char¬ 
acteristic functions, then one of the solutions of the 
boundary problem will also satisfy the initial condi¬ 
tions. 

This result can be generalized. If the initial distri¬ 
bution of the pressure is a linear combination of 
several of the characteristic functions i p k (y), then we 
shall show that these initial conditions can be satis¬ 
fied by a corresponding sum of normal modes. Sup¬ 
pose 

p(y,0) = 2] (152) 

j 

where the D y are any constants. Then the distribu¬ 
tion of pressure given by 

p(y,t ) = 2Z A j cos 2irfj(t - tj}\pj{y) (153) 

j 

will satisfy the initial conditions (152) if only the Ay 
and 6 y are chosen so that 

Ay COS 2x/yey = Dy. (154) 

The expression (153) also satisfies the wave equation 
and the boundary conditions since it is a sum of 
normal modes; hence, it is the solution to the problem 
when the initial pressure disturbance can be expressed 
as a finite sum of characteristic functions. 

Now suppose the initial disturbance cannot be ex¬ 
pressed in the form of (152), but is a general function 



36 


WAVE ACOUSTICS 


f(y). It is a remarkable fact of mathematics that if 
we allow the sum to be an infinite one, then any func¬ 
tion of y can be expressed in that form. That is, it is 
possible to express f(y) as 


f(v) = I>*y(y) 

j'=i 


” . 2x (155) 

sin — y 

j= 1 K i 


where the c, depend on the law of formation of the X y 
[whether (145) or (149), etc.3, and usually become 
small very rapidly as j increases. 

If the boundary conditions are (139a), then the Xy 
are given by equation (145), and we have 


f(y) = sin ( 156 ) 

y=l L 

In equation (156) the coefficients c y are given, ac¬ 
cording to the usual laws of Fourier analysis, by 

C ‘ = fio ^ siU ^L dy ' ( 157 ) 


The c« are called Fourier coefficients of f(y). Once 
we know these Fourier coefficients, we can solve our 
problem as in the finite case. We have 

p(v, 0) = Ylcjipj(y) (158) 

;'=i 

as our initial condition; and 


Piyff) = Z -Ty COS 2irfj(t - ej)ti(y) 

j =i (lo9) 

A y COS 2x/y«y = Cy 

as the set of solutions to the total problem. 

While each term in the sum (159) represents a sta¬ 
tionary state of vibration, the infinite sum is not sta¬ 
tionary, in view of the fact that the terms have 
different frequencies /y. 


2.7.2 General Waves in a Medium 
with Parallel Plane Boundaries 

Section 2.7.1 showed how the assumption of a 
stationary state led to a possible solution which satis¬ 
fied the wave equation, the boundary conditions, and 
the initial conditions. However, the treatment in 
Section 2.7.1 was restricted to the case of plane waves 
moving perpendicular to two enclosing plane bound¬ 
aries. This section explains how this method may be 
generalized for the case of general waves in a medium 
with two parallel plane boundaries. 


We assume again a stationary state in the medium, 
of the form 

p(x,y,z,t) =cos 2 t Tft-\p(x,y,z). (160) 

If this solution is substituted into the wave equation 
(27), we find that \p satisfies a partial differential 
equation of the form 


<f\p d~\p d-\p 47T 

— + ^+ -^7 + —<A = 0 

dx~ dy dz- X“ 


(161) 


which is time independent. In addition, \p must 
satisfy the boundary conditions imposed at the 
bounding planes y = 0 and y = L. The boundary 
conditions may be of the form (139a), (139b), or 
(139c). 

We shall treat only the case characterized by the 
conditions (139a). The treatment of the other cases 
is completely analogous. We attempt to find a solu¬ 
tion of equation (161) of the form 

i(x,y,z) = sin ^G(x,z) (162) 

A u 

in which the constant X,, must have one of the values 
2 L 


X„ = 


J 


j = 1, 2, 3, 


(163) 


9 2 G d 2 G J 1 l\ 

+ a? + Mx- - xlZ “ a (164) 


to satisfy the boundary conditions. For the G{x,z) 
we have the equation 
6H 
dx 2 

Any solution of this equation when multiplied by 
cos 2-rrft sin 2iry/\ y is a solution of the wave equation 
(27) and also satisfies the boundary conditions (139a). 
Equation (164) will be satisfied by any plane wave 
solution in the xz plane with the wavelength X* 
given by 

1 = ]/- - —• 

X* f X 2 X* 

Such a solution will be of the form 

2 tv 2tt 

G = Oi cos —s -f ar sin — s :s = x cos 6 + 2 sin 6 ■ 

X* X* 

(164a) 

It is easily verified that this function G satisfies (164). 
If Xy < X, X* is imaginary instead of being real. In 
that case, the solution should be written in the form 


G = b\e 


(2ir/X> 


T 60 C 


-(2r /X')s 


in which X' is the real constant given by 






NORMAL MODE THEORY 


37 


G is thus the sum of two terms, one increasing ex¬ 
ponentially with the distance s from the source, the 
other decreasing exponentially. The only solutions of 
this character which have physical significance are 
those for which the first (increasing) term is zero. If 
5i were not zero, the sound intensity would increase 
rapidly with the distance from the source and that is 
physically impossible. 

Since the greatest possible value of \ u is 2L, or 
twice the depth, it follows that sound of wavelength 
greater than 2 L will have an exponential rather than 
a trigonometric solution; because bi must be zero, 
such sound will suffer an exponential pressure decay 
with increasing range. It is clear that the longer the 
wavelength, the more rapid will be the decay. A more 
detailed discussion of this type of transmission is 
given in a report by the Naval Research Laboratory 
[NRLJ, where the detailed properties of the bottom 
are taken into account. 4 

The angle 6 in the solution (164a) may be chosen 
arbitrarily. Thus to any value of j in the equation 
(163) belongs an infinite set of characteristic func¬ 
tions. These characteristic functions can be com¬ 
bined to satisfy particular initial conditions; how¬ 
ever, the rules for their combination are too involved 
to be presented here. 

The derivation of the wave equation (27) was based 
partly on the assumption that the velocity of propa¬ 
gation was everywhere the same, in other words, that 
the medium w'as homogeneous. Let c be an arbitrary 
function of ( x,y,z ). In the ocean, the variation in 
sound velocity is due mainly to the variation in water 
temperature with depth. 

To assume that the velocity is variable, amounts 
for most practical purposes to assuming that c in 
equation (27) is now a function of position. The 
method of normal modes can be applied to find a 
solution in that case just as in the case of constant 
sound velocity. As before, we assume a stationary 
state of form (160). Substituting equation (160) into 
the wave equation, we get as the time-independent 
differential equation the following: 


w + w + w ML 0 

dx 2 dy 2 dz 2 c"(xyz) 


(165) 


which differs from equation (161) only in that c is 
now variable. The solution of equation (165) satis¬ 
fying the imposed initial and boundary conditions 
can be found as before by the superposition of an 
infinite number of normal modes; in this case of vari¬ 
able c, however, the computation of the characteristic 


values and functions is more troublesome. An ap¬ 
plication of this type of analysis is discussed in 
Section 3.7. 


2.7.3 Intensity as a Function of 
Phase Distribution 

Whenever the sound source is harmonic, the pres¬ 
sure distribution resulting from given initial and 
boundary conditions can be written in the form 

p = a(x,y,z)e 2 * im - t{x ' v '° n (166) 

a and e being real functions of x,y, and z. For some 
purposes it is convenient to set 

V(x,y,z) 


t{x,y,z) = 

Co 

so that equation (166) becomes 

p = a(x,y,z)e 27rifLt - {VM1 
or, explicitly for the real pressure, 

<■ - 3 


p = a(x,y,z ) cos 2nf\ 


(167) 


(168) 


(169) 


Since we know from Section 2.4.2 that I = pu, we 
must derive an expression for pu. This is done by 
making use of equation (44), relating the derivatives 
of p and u x : 

du x 1 dp 

dt p dx 

From equation (169), then 


(170) 


where 


dp ( ■ dV i da u 

— = a (sin H) -1-cos H 

dx Co dx dx 

V 

Co 


H = 2tt/1 t 


(171) 


Therefore, from equation (170), 
du x 

dt p 

Integrating this, we get 


a . 2tt/ dV 1 da 

- sin H ---cos H —- 

Co dx p dx 


Cl dV 

u x = — cos H — 

pCo dx 


1 • u da 

——- sin H - 

2wfp dx 


(172) 


From equations (172) and (169), we obtain 

a 2 r dV a . TT rr da 

pu x - — cos- H -—— sin H cos H — (1/3) 

pCo dx 2irfp dx 

The average energy flow in the x direction /*, at the 
point x,y,z, is just the time average of equation (173) 
over a complete period. The time average of the 





38 


WAVE ACOUSTICS 


square of the cosine is the time average of the 
product of the sine and cosine is zero. Thus, 


Similarly, 



(174) 


Since I = (/* + /* + /*)’, we have the following ex¬ 
pression for the intensity 


1 = 


2pco 




(175) 


The relations (174) and (175) will be found useful 
in Chapter 3 when the equivalence of wave acoustics 
and ray acoustics is investigated. 


2.8 PRINCIPLE OF RECIPROCITY 

The principle of reciprocity makes a statement con¬ 
cerning the interchangeability of source and receiver. 
Very crudely, the import of the statement is that if 
in a given situation the locations and orientations of 
source and receiver are interchanged, the sound pres¬ 
sure measured at the receiver will be the same as be¬ 
fore. To be true, under the most general conditions, 
this statement has to be qualified in detail. The fol¬ 
lowing is an attempt to formulate the General 
Reciprocity Principle. 

Assume that a source of a given directivity pattern 
b and a receiver of a different directivity pattern b' are 
placed in a medium with a particular distribution of 
sound velocity c(x,y,z), enclosed by boundaries of 
any given shape with any particular boundary con¬ 
ditions; let the output of the source on its axis be 
given by an amplitude A at one yard. The receiver 
will then record some pressure amplitude, correspond¬ 
ing to the amplitude B on its axis. Now let the source 
be replaced by a receiver having the same orientation 
of its axis and having the directivity pattern b'; 
assume also that the receiver is replaced by a source 
which has the same orientation of its axis, the direc¬ 
tivity pattern b, and the output A on its axis. Then 
the new receiver will again register a pressure 
equivalent to that of a sound wave incident on its 
axis with an amplitude B. 

The proof of this theorem is difficult in the general 
case, and will not be reproduced here. Instead, \\e 
shall give the exact proof for the simple case of a plane 
source emitting plane waves into a medium that 


satisfies boundary conditions of the type (139). We 
shall then indicate, roughly, how the proof can be 
generalized. 

Suppose the pressure is a function of y and t only. 
The medium may be inhomogeneous, but both the 
density and bulk modulus are assumed to be a func¬ 
tion of y only. Then, the sound velocity will be a 
function of y. The wave equation for this case there¬ 
fore reduces to 


d 2 P v d' 2 P 

dt°- dy 2 

and its solution, by equation (100), will be 


(176) 


p(y,t) = 'P(y) cos 2-irft (177) 


where \p{y) is obtained from 
d~\I/ 

t ©; + k 2 (y)\p - 0, k 2 = 

dy- 


4 IT 2 / 2 

c-(y) 


47r 2 

X 2 (y) 


(178) 


We assume the medium is bounded by the planes 
y = 0 and y = L and satisfies boundary conditions 
of the type (139a), (139b), or (139c). 

We assume that the plane y = a is a source of 
sound. If by ip a (y) is meant the function defining the 
pressure amplitudes at every point of the medium, in¬ 
cluding near y = a, then i p a satisfies equation (178) 
everywhere except near y = a. That is, it satisfies 


d 2 \p a 

dy 1 


+ k 2 (y)ta = A(y ) 


(179) 


where A (y) is very large in the immediate neighbor¬ 
hood of y = a, and zero everywhere else. Then the 
magnitude of the plane source S n , located at y = a, 
will be defined by 

Sa = £ A(y)dy = A(y)dy (180) 


where it is understood that S a is the limit of the 
integral as 5 approaches zero. 

In the same way, we define a plane source at 
y = b by assuming an amplitude function \pb(;y) 
which satisfies equation (178) everywhere except 
near y = b, that is, it satisfies 

“ + k 2 (y)^b = B(y) (181) 

dy- 


where B(y) is very large in the immediate neighbor¬ 
hood of y = b, and zero everywhere else. Then the 
magnitude of the source at y — b will be 

■'b+i 


s h = 


I 


B(y)dy. 


(182) 





INADEQUACY OF WAVE ACOUSTICS 


39 


By multiplying equation (181) by \p a , and equation 
(179) by t p b , and by subtracting the latter result from 
the former, we get 

d 2 d/ b d 2 \f/ a 

tc— - ib— = t a B - \f/ b A 

dy- dy- 

which may be rewritten as 

d ( d\J/ b d\f/ a \ 

iX+% - - **■ (183) 


Equation (183) holds if \p n and \p b are arbitrary 
functions of y, and A and B are defined by equations 
(179), (181). We can integrate (183) over the entire 
extension of the fluid between y = 0 and y = L, and 
get 

(*s? - ** X ” i> - (184) 


Since \p a and \J/ b each satisfy some combination of 
\j/ = 0 or d\[//dy = 0 at y = 0, and y = L, the left- 
hand side vanishes identically, and 
•x, 

(<A„B - 4'bA)dy = 0. (185) 


f 

Jo 


Equation (185) is valid for all functions of \j/ a and \f/ b 
and satisfies equations (179), (181), and the boundary 
conditions (139). 

Since B = 0 except near y = b, by equation (181), 

J -»L fb+S 

\paBdy = ta(b) I Bdy = \J/ a (b)S b 
0 Jb-S 

because of equation (182). Similarly, 

J 'L /•a+S 

foAdy = i b {a) I Ady = t b (a)S a . 

0 Ja-S 


Therefore, equation (185) becomes 

S„Mb) ~ Sjh(a) = 0. (186) 

If both sources are of equal magnitude, then S a = S b , 
and 

Mb) = Ma). (187) 

That is, if two plane sources of equal strength are 
emitting plane waves into a “stratified” medium, 
where the sound velocity obeys an arbitrary law, 
and where boundary conditions are of the form of 
(139), then the first source (at y = a) produces at 
y = b the same acoustic pressure which the source 
at y = b would produce at y = a. 

It is interesting to note that we have proved equa¬ 
tion (187) without solving explicitly for the pressure 
amplitudes. We remember that even in this one¬ 


dimensional problem the equation (178) with bound¬ 
ary conditions usually cannot be solved for ^ in terms 
of elementary functions if the sound velocity is an 
arbitrary function of y. However, we found we could 
prove equation (187) merely by assuming that \p a and 
^6 were solutions of the wave equation with initial 
and boundary conditions, and by following up the 
consequences of that assumption. 

In the general case, in which we no longer assume 
perpendicular incidence on plane parallel boundaries, 
the proof is more complex. Instead of equation (178), 
we have the more complicated form of (165). Equa¬ 
tions (179) and (181) must be replaced by equations 
with the same left-hand sides as equation (165), but 
with right-hand sides which are different from zero 
only in the immediate vicinity of a particular point, 
( x a ,y at z a ) and (x b ,y b ,z b )> respectively. The distribution 
of the functions A and B about these two points de¬ 
termines the directivity of the source considered. 

The integration (184) must be replaced by a volume 
integral, or rather, by an infinite series of volume 
integrals to account fully for the two directivity 
patterns, the left-hand sides of which can be shown 
to vanish. From there on, the proof runs similarly to 
the plane case. 

The foregoing remarks have applied only to the 
case of propagation in a perfect fluid. It can be shown 
that the reciprocity principle holds, with additional 
qualifications, for propagation in a viscous fluid also. 


2.9 INADEQUACY OF WAVE ACOUSTICS 

In this chapter, we have set up a schematic picture 
of the transmission of sound in the ocean, and pro¬ 
ceeded to derive a rigorous mathematical description 
of our schematic picture. Unfortunately, the results 
obtained cannot be used directly as a basis for the 
prediction of the performance of sonar gear. The 
schematic picture is not nearly complete enough from 
a purely physical point of view; furthermore, even 
the simplified schematic picture can be solved rigor¬ 
ously only for simple cases; and in the cases where 
solutions are possible, the calculations are very 
difficult. 

The physical picture is inadequate on several 
counts. For one thing, boundary conditions like p = 0 
of dp/dy = 0 at the boundaries are only a vague 
description of what actually happens at the bound¬ 
aries. The surface is not a perfect plane, but is usually 
disturbed and uneven, with the result that even plane 
waves are not reflected according to the law of reflec- 



40 


WAVE ACOUSTICS 


tion; they are partly reflected in a direction depend¬ 
ing on the direction of the surface and also partly 
scattered in all directions. Neither does the bottom 
obey the postulated conditions; it is never infinitely 
dense; at best it is rocky; at worst it is so muddy 
that it can hardly be called a boundary. The medium 
itself, the sea water, is not completely described by 
its density and its bidk modulus. There are many 
inhomogeneities in the sea volume, such as bubbles, 
floating plant and animal life, fish, and others. For 
all we know, such inhomogeneities may produce a 
very important part of the observed transmission 
loss, perhaps as important a part as the variations in 
sound velocity. 

The mathematical difficulties should be apparent 
to anyone who has even glanced at the remainder of 
the chapter. Even when the boundary conditions can 
be formulated exactly, and initial conditions are 
simple, the exact solution of the problem usually can¬ 
not be presented. In the general case, it can be proved 
that a solution exists and is unique, but the solution 
cannot be written in a formula which would provide 
a practical basis for intensity calculations. The 
primary benefit of the rigorous approach is that one 
can derive certain very useful properties of the sound 
field, such as the principle of reciprocity and the de¬ 
pendence of intensity on the phase distribution, with¬ 
out going into the exact solution itself. 


How, then, are we going to predict the sound field 
intensity? We certainly cannot go out and measure 
the intensity in all cases; such measurements are 
time-consuming, and provide information only about 
that particular part of the ocean at that particular 
time. We should have some method for estimating 
the intensity field, at least qualitatively, so that the 
observed intensity data can at least be interpreted 
according to a frame of reference; mere data without 
some reference to a theoretical scheme are useless. 

In the theory of light, this problem was solved by 
using the methods of ray optics. The fundamental 
problems about optical instruments, like those for tele¬ 
scopes, can be solved by ray tracing methods without 
resorting to the exact solution of the wave equation. 
This ray theory is based on the assumption that light 
energy is transmitted along curved paths, called rays, 
which are straight lines in all parts of the medium 
where the velocity of light is constant, and which 
curve according to certain rules in parts where the 
velocity of light is changing. This light-ray theory is 
valid in all cases where obstacles and openings in the 
path of the radiation are much greater in size than 
the wavelength of light. 

In the next chapter, we shall describe the applica¬ 
tion of ray methods to underwater sound transmis¬ 
sion and shall also examine the validity of this type 
of approximation. 



Chapter 3 

RAY ACOUSTICS 


C hapter 2 was devoted to the rigorous computa¬ 
tion of the acoustic pressure p as a function of 
position in the fluid and of time. In situations where 
the acoustic pressure could be determined the sound 
intensity at an arbitrary spot and at an arbitrary 
time could be calculated. However, it was noted that, 
in most situations involving initial and boundary 
conditions similar to those met with in actual sound 
transmission in the ocean, this computation was at 
best very laborious and at worst completely impos¬ 
sible to carry out. Ray acoustics provides a more con¬ 
venient though less rigorous approach. 

In the study of sound the ray concept has not 
played so great a role as in optics. The reason for this 
is that the wavelengths of most audible sounds are 
not small compared to the obstacles in the path of the 
sound. Consequently, sounds audible to the ear do 
not travel straight-line or nearly straight-line paths; 
they bend around corners and fill almost all of any 
space into which they are directed. However, for the 
short wavelengths used in supersonic sound, the ray 
methods have an important, if limited, application. 
This chapter elaborates the theory of sound rays, 
describes the computation of sound intensity from the 
ray pattern, and finally, examines the conditions 
under which ray intensities may be expected to ap¬ 
proximate the intensities calculated according to the 
rigorous methods of the second chapter. 

3.1 WAVE FRONTS AND RAYS 

3.1.1 Spherical Waves 

The wave equation was solved explicitly for p in 
one very important case: an impulse sent out by a 
point source into a homogeneous medium under the 
assumption of spherical symmetry. The pressure as a 
function of time and space was found to be 

(1) 

In this expression, r is the distance from the source, 


and the function /(< — r/c) is determined by the out¬ 
put of the source. Specifically, the source can be 
characterized by the statement that, while the pres¬ 
sure itself at the source is infinite, the product rp in 
the immediate vicinity of the source has a finite value 
at every instant, namely/(f). 

Obviously, this function /(f — r/c) determines 
when a pulse emitted by the source at a particular 
instant will arrive at a given point of space. If the 
pulse should, for instance, have started in time at 
some instant t = e so that for all values of the argu¬ 
ment less than € the function /(f) vanishes, then the 
onset of the disturbance at a distance r from the 
source will be observed at the time 

r 

t — « d- 

c 

Likewise, we find that the front of the pulse will have 
reached at the time f a distance r, given by 

r = c(t — «). (2) 

What has just been stated about the front of the 
pulse might just as well have been said about any 
other specified part of the pulse; only, e would in that 
event characterize the time at which the specified 
part of the pulse was radiated by the source. What 
has been called, vaguely, part of the pulse, is often 
more concisely called phase, particularly in connec¬ 
tion with harmonic pulses. The term e then char¬ 
acterizes the phase of the pulse considered and is 
ordinarily referred to as a phase constant. 

The surface defined by equation (2) is, of course, a 
sphere of radius c(f — e) at the time f. As the time 
increases, the radius of the sphere increases at the 
rate of c units per second. The surface of this sphere 
of constant phase is called the wavefront. The energy 
represented by the disturbance of equilibrium condi¬ 
tions clearly spreads out radially from the source. We 
may focus attention on the direction of energy flow 
by mentally drawing an infinite number of radii from 
the source to the wave front. These radii may be re¬ 
garded as representing the paths of energy flow and 


41 


42 


KAY ACOUSTICS 


may be called sound rays, in analogy with light rays. 
Sound energy may be regarded as traveling out along 
these rays with the speed c. The wave fronts assume 
in this description the secondary role of surfaces 
everywhere normal to the rays. 

An individual sound ray cannot exist as a physical 
phenomenon. An isolated sound ray would mean a 
state of the fluid where the condensation was confined 
to the immediate neighborhood of a particular 
straight line. Beams of narrow cross section can be 
produced by directing a wave front onto a very nar¬ 
row slit; but if the size of the slit becomes comparable 
to the wavelength of the sound, the sound leaving the 
slit is not a narrow beam, but a cone. This phenome¬ 
non is called diffraction, and will be discussed in 
Section 3.7. It is mentioned here only to indicate 
that the concept of a sound ray refers not to the 
propagation of a narrow beam with sharp edges, but 
merely to the direction of propagation of actual wave 
fronts. 

Spherical wave fronts represent only one particular 
case of sound propagation, even in an infinite fluid. 
First, the wave front can be spherical only if the 
initial disturbance has no preferential direction (a 
vibrating bubble satisfies this condition). Second, the 
expanding surface remains a sphere concentric with 
the origin only if the sound velocity c is either con¬ 
stant throughout the fluid, or has spherical symmetry 
about the sound source. 

The wave factor/(f — r/c) in equation (1) is re¬ 
sponsible for the conclusion that the disturbance is 
propagated with the velocity c. The remaining factor, 
1/r, is called the amplitude factor since it is responsi¬ 
ble for the decrease in sound intensity as the distance 
from the source increases. The rate at which sound 
intensity is weakened with distance can be easily 
computed by using the concept of rays as carriers of 
sound energy, provided we assume that energy is 
generated only at the source and then flows through 
space without gain or loss. For reasons of symmetry, 
the energy flow from the source must take place 
along the radial sound rays. There will be a definite 
number of rays inside a unit solid angle. These rays 
will intercept an area of 1 sq ft on a sphere of 
radius 1 ft whose center is at the source, an area 
of 4 sq ft on a sphere of radius 2 ft, and generally 
r 2 sq ft on a sphere of radius r. Since the total energy 
flow is the same for all these spherical surfaces, the 
energy flow per unit area, or sound intensity, must 
be inversely proportional to the square of the distance 
of the unit area from the source. 


3.1.2 General Waves 


The frontal attack on the wave equation was the 
solution of the boundary problem by the method of 
normal modes. This method was found to be too com¬ 
plicated. It was shown in Section 3.1.1 that the 
method of sound rays gave a simple and plausible ac¬ 
count of sound propagation for the case of spherical 
symmetry. A natural approach to the general prob¬ 
lem would be to generalize the definition of sound 
rays, and see if light could thereby be thrown on the 
general case of variable sound velocity and arbitrary 
initial distributions of p. 

First we must generalize the definition of wave 
fronts. In what follows we shall restrict ourselves to 
harmonic sound waves, that is, sound waves which 
have been produced by a sound source which under¬ 
goes single-frequency harmonic vibrations. In ac¬ 
cordance with Section 2.4.3, the pressure at any 
point inside the fluid can be represented as the real 
part of an expression having the form 

V = A(x,y,z)e i9(M) (3) 

in which the angle d at each point in space increases 
linearly with time, 

d = 2irf[t - t{x,y,z)~\. (4) 

We shall now call a wave front all those points at 
which the phase angle 6 has a specified value, say d». 
At any time t, this surface is defined by the equation 


e(x,y,z) = t 


2nf 


(5) 


For later convenience, we shall replace t(x,y,z) by an 
expression W(x,y,z)/co, in which Co is the velocity of 
sound under certain designated standard conditions. 
Equation (3) then takes the form 

p = A(x,y,z) e 2 ’ r,/ C ( -' V( f 0 i/ " ) ], (6) 


both A and W being real functions of the space co¬ 
ordinates. The defining equation (5) of an individual 
wave front assumes the form 


W{x,y,z) = co(t - < 0 ) (7) 

, , 

where t 0 = —-■ 

ZirJ 

The term t 0 has different values for different wave 
fronts, but is constant both in space and in time for a 
given wave front. The function W clearly has the 
dimension of a length. 

In order to make use of the concept of sound rays 
to describe the energy propagated by such generalized 





FUNDAMENTAL EQUATIONS 


43 


wave fronts, we must also generalize the definition of 
a ray. We can no longer assume that the rays are 
straight lines since we concede the possibility of re¬ 
fraction and reflection. We shall, however, retain the 
property that the rays are everywhere perpendicular 
to the wave fronts. It is, of course, by no means 
obvious that the results of this new approach will 
agree with results from a direct solution of the wave 
equation plus initial and boundary conditions. A 
comparison between the results from the ray pattern 
approach and the results from a rigorous treatment 
of the wave equation will be carried out in Section 3.6 
once the ray method has been fully described. It will 
be found that in many practical situations these two 
approaches lead to similar results. 

Geometrically, the rays and successive wave fronts 
can be constructed as in Figure 1. The wave front at 

w=c dt 



Figure 1 . Huyghens’ method for constructing suc¬ 
cessive wave fronts. 


time t = 0 (whose equation is given by W = — co<o) 
is first drawn. In order to determine the wave front 
at the time dt, the small ray elements are drawn as 
straight-line segments perpendicular to the initial 
wave front, as at (xi,yi,Zi). In the time dt, the end 
point of the ray starting at (xi,i/i,Zi) will have pro¬ 
gressed to a point a distance cdt from the initial 
wave front, where c is the velocity at the point 
(xi,yi,Zi). If this process is carried through for all the 
points on the initial wave surface, the end points of 
all the small ray elements will determine a second 
surface, which may be regarded as the wave front at 


the time dt. By performing this process many times, 
the wave front can be obtained at any time t. This 
method of determining wave fronts by gradually 
widening an initial wave front was first suggested by 
the Dutch physicist, Huyghens, in the seventeenth 
century, for the solution of problems in optics. 


3.2 FUNDAMENTAL EQUATIONS 

3.2.1 Differential Equation of the 
Vi ave Fronts 

Since the construction of wave fronts described in 
the preceding section is purely geometrical, it must 
be reformulated in mathematical terms for use in an 
algebraic analysis of the sort we are carrying out. 



Figure 2. Differential ray path. 


Let P in Figure 2 be any point on the wave front at 
time t. The equation of the wave front is given by 
equation ( 7 ). Let the coordinates of P be ( x,y,z ); let 
PP' be the ray element emanating from P at the end 
of a time interval dt; and let a,P,y be the direction 
cosines of PP'. Then the coordinates of P' are 
(x + acdt, y + /? cdt, z + ycdt). Further, the wave 
front at the time t + dt is given by the equation 

W(x + acdt, y + /3 cdt, z + ycdt) = Co(t — to + dt). (8) 


If cdt is assumed to be very small, the left-hand side 
of equation (3) is very nearly equal to 

/ dW dW dPF\ 

W(x, V ,z) + {a— + 0— + y—)cdt. 

If we substitute this expression into equation (8), and 
use equation (7), equation (8) reduces to 


dW dW dW co 

a — ^ f- y—— = - • 

dx dy dz c 


(9) 


The direction cosines a,(3,y will next be eliminated 
from equation (9). It is a well-known theorem 1 of 
analytical geometry that the direction cosines of the 
normal to the surface W = constant at the point 
( x,y,z) satisfy the proportion 

dW dW dW 
dy dz 


a :y = — 
dx 










44 


RAY ACOUSTICS 


Because the sum of the squares of the direction 
cosines equals unity, 

a 2 + /3 2 + 7 2 = 1, 


the constant of proportionality in the multiple pro¬ 
portion above can be determined, and we obtain 


a 

0 



-*c W 

dx (10) 
~*dW 


J dy 


, etc. 


By substituting these values of a,(i,y into equation 
(9) and squaring both sides, 


/ dW 
\ dx 


airV 

) + 


airy 

dy) 


+ — 


(dW\ = cl 
\dz ) c-{x,y,z) 


If we define n, the index of refraction, by 


(HI 


n(x,y,z) = 


c{x,y,z) ’ 


equation (11) becomes 


©' 




(?) = B ' (w) - 


12 ) 


(13) 


Equation (13), often called the eikonal equation, is 
the fundamental equation of ray acoustics. It is a 
partial differential equation satisfied by all functions 
W which can define wave fronts according to equa¬ 
tion (7). Initial conditions for equation (13) are 
usually of the form that IT has the value zero for all 
points ( x,y,z ) on a particular surface. 

Once the solution IT of equation (13) has been 
found, the ray pattern can easily be drawn. The direc¬ 
tion cosines of the rays at every point of space can 
be computed from equation (10); more simply, if 
equations (10) and (13) are combined, 


1 dW 1 dW 1 dW 

^ > y - 

n dx n dy n dz 


(14) 


Later we shall eliminate the function W from equa¬ 
tion (14), and derive a set of ordinary differential 
equations, which together determine the course of 
each individual ray. First, however, we shall give a 
simple example illustrating how the ray pattern may 
be calculated from the partial differential equation 
(13) for the wave fronts. 

Let us consider the special case where the sound 
velocity c depends only on the vertical depth co¬ 
ordinate y. Thus, the sound velocity is assumed 
constant everywhere on a particular horizontal plane. 
We shall examine only the ray pattern in one vertical 


plane, which we can take as the xy plane. Then equa¬ 
tion (13) reduces to 



To find a simple solution of equation (15), we 
assume that W(x,y) is the sum of a function of x and 
a function of y. 

W{x,y) = W 1 (x) + Wfy). 

Substituting this expression into equation (15), we 
obtain 



To obtain a family of solutions, we put dWi/dx = k, 
where k is an arbitrary constant. Then, the differ¬ 
ential equation will be satisfied if 


dW, 

dy 


= V n-{y) — k 2 ; W 2 = f V n 2 (y) - k 2 dy. 

J o 


Therefore, in view of the assumed nature of IT, the 
equation (15) will be satisfied by all functions IT de¬ 
fined by 

IT (x,y) = kx + y/n 2 {y) - k 2 dy (16) 


where k is any constant. A particular choice of k 
corresponds to a particular solution W(x,y) and 
therefore to a particular set of wave fronts (7). 

The direction cosines of the rays, corresponding to 
this choice of k, can be calculated from equations 
(14) and (16). 



We use these expressions to obtain the equations 
y = y(x) of the sound rays. If we denote by dy/dx 
the slope of the direction of the ray at the point 
(x,y), then 

dy 


dx 


_ ]/n 2 (y) 


- 1. 


This equation integrates immediately to 
r dy 

X = J q _ ^ + To 


n-{y) 

k 2 


- 1 


(17) 


where x 0 is an arbitrary constant. Regard the k as 
fixed, and the x 0 as variable; then equation (17) gives 
an infinite set of curves which satisfy the definition 
of rays. 
















FUNDAMENTAL EQUATIONS 


45 


If n is a constant, that is, if the sound velocity 
is independent of depth, the rays (17) are clearly 
straight lines. 

3.2.2 Differential Equations of Rays 

It may be argued that the replacement of the wave 
equation by the ray treatment as represented by the 
differential equation (13) has little to recommend it¬ 
self. It appears that one difficult partial differential 
equation has merely been replaced by another, which 
might resist attempts at solution as effectively as the 
first one. 



Figure 3. Specification of direction of ray element ds 
by direction cosines. 


Further examination shows, however, that the new 
equation (13) has two properties which tend to sim¬ 
plify its solution. First, equation (13) contains no 
time derivatives. This means that it describes the 
propagation of a disturbance in terms independent 
of the frequencies which make up this disturbance. 
Second, it is possible to set up ordinary differential 
equations that describe the path of individual rays; 
the latter equations will be derived in this section. 

We start with the equations (14), from which we 
proceed to eliminate W. This can easily be done by 
use of the formulas d 2 W / dxdy = d 2 W/dydx, etc. By 
differentiating the first equation of (14) with respect 
to y, the second with respect to x, and by equating 
the results, a relation between a, P, and n is obtained. 
Proceeding similarly with the other equations, we 
obtain the following relationships which must hold 
at any point of the ray pattern: 


d(na) d(ni3) d(na) d(ny) d(nP) d(ny) 

dy dx ’ dz dx dz dy 

( 18 ) 

These equations can be developed further to yield 
the changes of a, P, and y along the path of an indi¬ 
vidual ray. If the arc length along the ray path from 
a given starting point is denoted by s, we have 


d(na ) 


d(na) dx ^ d(na ) dy d(na ) dz 


ds 


dz ds 


ds dx ds ' dy 
We see from Figure 3 that 

dx dy dz 

ds a ’ ds ^ ’ ds 

Thus equation (19) turns into 

d(na ) d(na ) d(na) d(na) 

= a ~ r P ~— + 7—~— , 

ds dx dy dz 

which, upon using the relations (18), becomes 


T- ( 19 ) 


( 20 ) 


d(na ) d(na) 

= a— -b 


ds 


dx 


d(na ) d(n<x) 

+ 7 - 


dx 


dx 


. dn 

= (a + P 2 + 7 2 ) — 
dx 


+ 


( da dp dy\ 

V'S + 'S+ ’'£>• (21) 


The first parenthesis equals unity, because it is the 
sum of squares of direction cosines; while the second 
parenthesis, which is equal to one-half times the 
derivative of the first one, vanishes. Thus equation 
(21) simplifies to 

d(na ) dn 

ds dx 


After similar calculations are carried out for d(np)/ds 
and d(ny)/ds, we get the following set of three ordi¬ 
nary differential equations: 

d(na ) dn d(nP) dn d{ny) dn 

ds dx ’ ds dy ’ ds dz 

It is understood that n, the index of refraction, is a 
given function of x,y,z. 

We now deduce an important result for the special 
case where the sound velocity is a function of the 
vertical depth coordinate y alone. We shall show that 
for this case the entire path of an individual ray lies 
in a plane determined by the vertical line through 
the projector and the initial direction of the ray. 

Let the origin of coordinates be taken at the pro¬ 
jector, and let the direction cosines of a ray leaving 



























46 


RAY ACOUSTICS 


the projector be a 0 ,/3o,7o, as in Figure 4. Since n de¬ 
pends only on y, equations (22) simplify to 

d(na) n d(np) dn d(ny) n /00 , 

HT ~ ° '• ~dT ~ dy ! nr ~ “• { ’ 
Thus, along any individual ray we have not = con¬ 
stant, ny = constant, which in turn implies 

7 

- = constant = k 
a 

along the ray. Then, the initial direction of the ray 
is (ao,/3o,Kao). 



Figure 4. Change of ray direction between point 

(0, 0, 0 ) and point (x, y, z). 

The direction at a general point P along the ray 
will be characterized by the direction cosines a,P,Ka 
because of the equations (23). It can easily be shown 
by the methods of analytical geometry that the nor¬ 
mal to the plane determined by OY (direction cosines 
0,1,0) and OA (direction cosines a 0 ,p 0 ,Kao) has the 
direction cosines k/V <r + 1,0, -1/Vr -f- 1. The 
direction of the ray at P is characterized by the direc¬ 
tion cosines a,(i,Ka; thus the ray direction at P is 
perpendicular to the normal to the plane AOY; hence 
the segment PB lies in that plane. Since P was any 
point on the ray, the entire ray must lie in the plane 
AOY. 

3.3 RAY PATHS FOR VERTICAL 
VELOCITY GRADIENTS 

3.3.1 Derivation of the Equations 
of Ray Paths 

We now solve the equations (22) for the special 
case where the sound velocity depends only on the 
vertical depth coordinate y and discuss this solution 


in detail. It is intuitively obvious that if we carry 
through the solution for the xy plane, then the ray 
pattern in any other plane through the vertical (y) 
axis will be identical in size and shape. 

Since the water depth increases in the downward 
direction, we shall take the y axis positive downward. 
We shall denote the angle which a direction in the xy 
plane makes with the positive x direction by 9, as in 
Figure 5. To avoid ambiguity, we must specify care- 



Figure 5. Change in rav direction over ray element 
PP'. 


fully the sign of the angle 9. We shall be concerned 
only with rays moving in the direction of increasing 
x, in other words, to the right in the figure. If the ray 
is gaining depth with increasing range, we give the 
angle 9 a positive sign; while if the ray is losing depth 
with increasing range, we give 9 a negative sign. These 
conventions, illustrated in Figure 6, enable us to use 



Figure 6. Conventions fixing sign of 6. 


the following relations both for climbing and de¬ 
scending rays: 

a = cos 9 ; /3 = sin 9 ; 7 = 0. (24) 

Since the sound velocity is assumed to depend 
only on y, we have 

dn dn 















RAY PATHS FOR VERTICAL VELOCITY GRADIENTS 


47 


and by reason of relations (24) the equations (22) re¬ 
duce to 

d(n cos 0) d(n sin 0) dn 

~di-=Ty- (25) 

From the first equations it follows that n cos 0 has 
a constant value along a particular single ray. That 
is, if P and P' are two points on the ray, then 


c° „ Co 
— cos 0 = — COS 0 . 
c c 


If, in particular, P is located at the depth where 
c(y) = Co, and if 0 O is the direction of the ray at this 
point, this equation becomes 

cos 0 c 1 

-T = - = - • (26) 

cos 0 O Co n 


Equation (26) is identical in form with Snell’s law in 
optics. 

The second equation in (24) is used to compute the 
curvature of the ray at any point. The curvature of a 
curve at a point on it is defined as dd/ds, the angle 
through which the tangent turns as one travels along 
the curve for unit distance. Because of our conven¬ 
tions for the sign of the direction angle 0, upward 
bending is always associated with negative curvature, 
and downward bending with positive curvature. 

From the relations (25), we have 


dn 

dy 


d(sin 0) . dn 

n ---b sin 0— 

ds ds 

d(sin 0) dd . dn dy 

n ---- + sin 0— — 

dd ds dy ds 

dd . dn 
n cos 0— + sin 2 0 — 
ds dy 


(27) 


since dy/ds = sin 0, from Figure 5. The solution of 
equation (27) for dd/ds yields 


dd 

ds 


1 dn 
ndy 


cos 0 = 


d(log n) 
dy 


cos 0. 


(28) 


Since log n = log Co — log c, equation (28) can be 
rewritten as 


dd 

ds 


d(log c) 
dy 


cos 0. 


(29) 


We can use equation (29) to describe, qualitatively, 
what happens when a ray travels to a layer just above 
it (dy < 0) of different sound velocity. If the new 
layer has higher sound velocity, the curvature dd/ds 
has a positive sign, and the ray is bent downward. 
If the layer just above has lower sound velocity, the 
curvature dd/ds is negative, and the ray is bent up¬ 
ward. We get the opposite result if the ray is traveling 


to a layer just below it (dy > 0) of different sound 
velocity. Thus we can say, in general, that a ray en¬ 
tering a layer of higher sound velocity is bent away 
from the layer, and a ray entering a layer of lower 
sound velocity is bent into the layer. 

In the open ocean the vertical velocity gradient 
usually falls into one of two types, depending on the 
temperature-depth variation. If the temperature does 
not depend on the depth, the velocity is determined 
by the pressure, which increases with depth; there¬ 
fore, in such isothermal water the sound velocity in¬ 
creases gradually with depth, and sound rays should 
possess slight upward bending. Another common case 
has the temperature decreasing with depth. Since 
velocity is much more sensitive to changes in tem¬ 
perature than to changes in pressure, the velocity will 
also decrease with depth, and the sound rays will 
bend strongly downward. The water temperature 
rarely increases with depth; when it does, the sound 
rays are bent strongly upward. 

We shall now examine, quantitatively, the change 
of curvature along an individual ray, and derive cer¬ 
tain relationships between the range and depth 
reached at time t by a ray leaving the projector at a 
certain angle. Assume that the projector is situated 
at the depth where c = Co; thus the ray may be 
characterized by its initial angle 0 O at the projector. 
Because of equation (26), equation (29) becomes 
dd _ dc cos 0o 

7 7 

ds dy Co 

The advantage of the representation (30) is that it 
gives the curvature along a single ray as a function 
of dc/dy only, since 0 O is constant for that particular 
ray. 

We consider, in particular, the case where the 
velocity gradient has the constant value a; that is, 
c = Cq + ay, (31) 


if the origin of coordinates is taken at the projector. 
At all points on the ray, in view of equation (30), 
de _ _ocos9o (32) 

ds Co 


We see from equation (32) that the curvature is 
constant along the ray; this means that the ray must 
be an arc of a circle. As the radius of curvature is the 
reciprocal of the curvature dd/ds, the radius r of this 
circle must be given by 


Co 

a cos 0o 


(33) 


If a is positive, the curvature (32) is negative, and 















48 


RAY ACOUSTICS 


the circular arc bends upward; but if a is negative, 
the circular arc bends downward. 

We can determine the center of the circle defining 
the ray by a simple geometrical construction. Figure 
7 shows the path of a ray leaving the projector at the 



Figure 7. Geometrical construction of ray path. 


angle 0 O into a medium of constant negative velocity 
gradient. The center of the circle is obtained by fol¬ 
lowing the perpendicular to PQ down through the 
medium a distance Co/a cos 0 O . It is a simple conse¬ 
quence of the geometry of the situation that this 
center will lie on the horizontal line a distance Co/a 
below the projector. For, from the illustration, 
RO = (co/o)(cos 0i/cos 0 O ), and 0i clearly equals 0 O . 
Similarly, if the constant velocity gradient is positive 
so that the rays are bent upward, the centers of the 
defining circles lie on a horizontal line a distance Co/a 
above the projector. It is easily shown that the dashed 
horizontal “line of centers” in Figure 7 is at the depth 
where the velocity c would equal zero if the assumed 
linear gradient extended indefinitely. 

An approximate solution in the general case where 
c is an arbitrary function of y can be obtained by re¬ 
peated use of the solution for constant gradient. Even 
a complicated velocity-depth curve can be closely 
approximated, as in Figure 8, by dividing the depth 
interval into a relatively small number of segments 
in each of which the velocity is assumed to change 
linearly with depth. Within each layer the ray path 
is an arc of a circle; and the total ray path is a con¬ 
secutive series of such arcs. 


VELOCITY 



Figure 8. Approximating velocity-depth curve by a 
succession of linear gradients. 


In practice, the path of the ray cannot be conven¬ 
iently plotted as a sum of circular arcs because the 
horizontal ranges are much greater than the depths of 
interest, and therefore their scale must be contracted. 
Instead, the ray is usually traced by calculating the 
angles 0i and 0 2 at which it enters and leaves a given 
layer, and the horizontal distance it travels in the 
layer. This calculation is illustrated in Figure 9, 



Figure 9. Ray path in layer of linear gradient. 


where the top of the layer is at depth y u the bottom 
is at depth y 2 , and the thickness of the layer is h. 

The ray leaves the projector at an angle 0o, enters 
the layer at the angle 0i, and leaves the layer at the 
angle 0 2 . Then, by equation (26), 

c(yi) cos 0 q 
c 0 


0i = arc cos 


02 = arc cos 


(‘ 


c(y 2 ) cos 0 O 


(34) 


where c{y i) and c(y 2 ) are calculated from equation 
(31). 


















RAY PATHS FOR VERTICAL VELOCITY GRADIENTS 


49 


Consider now the chord PiP 2 converting the end 
points of the circular arc. The direction 0 of this 
chord is by simple plane geomet ry 1 2 ( 0 1 -f 0 2 ); and 
its length is therefore given by 


P1P2 


h 

sin |( 0 i + 0 2 ) 


(35) 


The increase in horizontal range due to the passage 
of the ray through the layer is P\P 2 cos 0 , or 

Range in layer = h cot §( 0 i + 0 2 ). (36) 

This result may be applied to the following prob¬ 
lem. Suppose we have a sum of layers of the sort 
shown in Figure 10; and we wish to find the horizontal 


VELOCITY 



Figure 10. Succession of linear gradients. 


range attained by the time the ray reaches the depth 
H below the projector. We let the bottom layer ex¬ 
tend just to the depth //; suppose this is the third 
layer below the projector. We know 0 O and we cal¬ 
culate 61 , 62,63 by the relations (34). Then the hori¬ 
zontal range to the depth H will be the sum of terms 
of the form (36): 

Horizontal range to H = fticot£(0 o + 0i) + 

h 2 cot ^(0i ~b 62 ) -f- /13 cot ^(02 -{- 63 ). (37) 

The inverse problem is a little more complicated. 
Suppose we wish to find the depth reached by a ray 
of initial direction 0 O by the time it has traveled a 
horizontal distance R in a stratified medium that 
consists of layers of thickness hi,h 2 ,h 3 , etc. We cal¬ 
culate the range Ri in the first layer, R 2 in the second 
layer, and so on, until the sum of these partial ranges 
is greater than R : 

Ri -f- Ro -b R 3 ^ R 
Ri -b R 2 "b R 3 ~b Ri > R- 

Then the depth the ray reaches at range R will be 


greater than hi + h 2 + h 3 and less than hi + h 2 + 
h 3 + hi. Its value may be obtained with sufficient 
accuracy by interpolation. 

The ray-tracing methods described in this section 
are too cumbersome to use in practice. A number of 
devices have been developed to facilitate the plotting 
of rays bent by known velocity gradients; these de¬ 
vices will be discussed in Section 3.5.1. 

3.3.2 Application to Depth Correction 

The ray-tracing methods described in Section 3.3.1 
have had a valuable application in correcting the 
depths determined by the use of tilting beam sonar 
gear. These instruments are used on surface vessels 
to determine the true depths of submerged sub¬ 
marines. They employ a transducer with good verti¬ 
cal directivity and tiltable in the vertical plane, which 
sends out echo-ranging pings at various angles of de¬ 
pression. When velocity gradients are absent, the 
sound rays are straight lines, and the true depth of 
the target is just the slant range times the sine of the 
angle of depression at the orientation for which the 
target returns the loudest echo. The depth finder 
computes this latter product automatically. When 
velocity gradients are present, however, this simple 
method often leads to serious underestimation of the 
target depth. In this section, we shall describe a 
method for estimating the error produced. 

For simplicity, we assume that the projector is at 
the surface, as in Figure 11. Let the apparent target 


PROJECTOR 



Figure 11. Error in target position due to refraction. 

angle be 0 O , and the apparent target depth indicated 
by the depth finder be F 0 ; the true depth of the target 
is designated by V. Our aim is to derive an expression 
for Y — To in terms of the way the sound velocity 
c varies with depth. 

Let y represent the actual depth attained by the 
sound ray at time t, and y 0 represent the apparent 
depth reached by the ray. Then 
yo = Co sin 6 0 t 


(38) 













50 


RAY ACOUSTICS 


where Co is the velocity of sound at the projector. 
Since the actual ray path is curved, all we can say is 
that y is some function of the surface velocity, the 
apparent angle 0 O , and the velocity-depth pattern. 
We can, however, give an exact expression for the 
increase in y during the time interval dt. If c is the 
sound velocity at the depth y, and 0 is the inclination 
of the ray at the depth y, we have 

dy = c sin ddt. (39) 

We now take differentials of both sides of equation 
(38), obtaining 

dy 0 = Co sin d 0 dt. (40) 

By dividing equation (40) by equation (39) to elimi¬ 
nate the time, 

dyo _ Co sin 0 O 
dy esin0 

We eliminate 0 from equation (41) by using Snell’s 
law (26): 

sin 6 = "\/1 — cos 2 6 = y/ 1 — [(c/co) cos 0 O ] 2 
so that equation (41) becomes 

dyo _ sin 9 0 _ 

The quantity c/c 0 represents the variation of velocity 
with depth. If e is defined by the relationship 

c 

- = 1 -f- €, 

Co 

then € represents the relative change in velocity as a 
function of depth. Rewriting equation (42) in terms 
of €, we obtain 

dyo _ sin 0 q _ 

dy (1 + e)[l — (1 + t) 2 cos 2 0 O ]‘ 

= { (1 + e) 2 esc 2 0 O [1 - (1 + c) 2 COS 2 0 O ]} (43) 

= [1+2(1- cot 2 0 O ) « + (1 - 5 cot 2 0 O ) c 2 
— 4 cot 2 0 O e 3 — cot 2 0 O « 4 ] -? 

upon multiplying out and collecting terms. 

Since percentage changes of sound velocity are 
always small in the sea, the quantity e is a very small 
fraction, almost always less than 0.02. Consequently, 
the terms with e 2 or higher powers of e in equation 
(43) may safely be neglected, giving approximately 

“ = [1 + 2e(l - cot 2 0 O )]“ ! . (44) 

dy 

If we define w by 

w = 2(1 — cot 2 0 O ), 


equation (44) may conveniently be rewritten as 
dy 


dyo = 


(1 + we)’- 


(45) 


It may be noted that although e is always much less 
than one, we is not necessarily so. We now integrate 
both sides of equation (45) between 0 and the true 
depth T, obtaining 


Y 0 



dy 

1 + We) 1 ' 


(46) 


The expression (46) provides a functional relation¬ 
ship between the true depth Y, the apparent depth 
l’o, and the velocity-depth variation e(y). In any 
practical situation it is possible to determine T 0 and 
0 O , and e{y) can be deduced from the temperature- 
depth variation indicated by the bathythermograph 
slide. Thus, all quantities in equation (46) are known 
except the true depth Y, which occurs only as the 
upper limit of integration. The value of Y may be 
estimated by using trial values for the upper limit 
of integration and by seeing which trial value yields 
a value for the definite integral closest to the known 
left-hand side T 0 . If the velocity-depth variation is 
not simple, the integrals must be evaluated by numer¬ 
ical integrations; but if e(y) is a linear function of y, 
or a succession of linear functions of y, the integrals 
can easily be evaluated exactly. 

Tables have been developed by the use of such 
methods for the depth errors expected in the presence 
of various types of velocity gradients. In preparing 
these tables it was assumed that the sound velocity 
versus depth curve could be approximated by judi¬ 
ciously chosen straight-line segments without intro¬ 
ducing too much error in the calculated depth error. 

Though equation (46) is the relation used in the 
construction of depth correction tables, it is interest¬ 
ing to carry the approximation two steps further. If 
we expand the integrand in powers of we and neglect 
all but the first two terms, we get 

To = J g (1 — iwe + • • •)dy 


which becomes 


Y - To 



+ terms in (we) 2 . 


To the same order of approximation, F 0 may be sub¬ 
stituted in place of Tas the upper limit of integration, 
which gives 



+ terms in (we) 2 . 


(47) 


t - y 











CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 


51 


When we is small the terms in (we) 2 may be neglected; 
under these same conditions Y — Y 0 is small com¬ 
pared with Y 0 . Equation (47) thus provides a useful 
approximation when the depth error is relatively 
small. By translating from e back to c this equation 
becomes 

w r 5 " 

Y ~ Y 0 ~ — I (c - c 0 )dy. (48) 

2c 0 J o 

The expression (48) has a simple interpretation in 
terms of the velocity-depth diagram. The integrand 
(c 0 — c)dy is just the black area in Figure 12; thus 



y 

Figure 12. Depth correction as area under bathyther¬ 
mograph trace. 


the integral from 0 to F 0 is the shaded area between 
the velocity-depth curve, the vertical line c = c 0 , and 
the horizontal line y = F 0 . Qualitatively, we may 
conclude that the depth correction will be large for 
steep gradients and larger if these steep gradients 
are located at shallow depth. 


3.4 CALCULATION OF SOUND INTENSITY 
FROM RAY PATTERN 

The foregoing sections were devoted exclusively to 
tracing the paths of individual rays and stated 
nothing about sound intensity in the ray pattern 
except for the special case of spherical waves. For 
that situation an assumption that the energy flows 
out radially along the sound rays led to the same 
inverse square law of intensity decay which was de¬ 
rived rigorously under “Spherical Waves” in Section 
2.4.2. It is a plausible generalization to assume that 
energy always travels out along the rays even when 
the sound velocity is not constant and the rays are 
curves. 


The assumption for the case of constant sound 
velocity and its generalization for the case of variable 
velocity are illustrated in Figure 13. In the left-hand 



SOUND VELOCITY CONSTANT WITH DEPTH 




SOUND VELOCITY CHANGING WITH DEPTH 

Figure 13. Effect of vertical velocity changes on ray 
paths. 


drawing, the rays are straight lines; and the energy 
radiated by the source into a small solid angle is con¬ 
fined inside the indicated cone. Because of this as¬ 
sumption, we get the exact inverse square law of in¬ 
tensity loss. In the right-hand drawing, the rays are 
curves; and the energy radiated by the source into 
the same small solid angle is confined inside the horn¬ 
shaped surface displayed. In this general situation 
the energy flow through normal unit area depends 
not only on the distance r from the source, but also 
on the total cross-sectional area of the horn which, in 
turn, depends on the way the rays are bent. Thus it 
is clear that the inverse square law will not, in general, 
be predicted even by this simplified ray treatment. 

3.4.1 General Formulas for Change 
of Intensity along a Ray 

The prediction of shadow zones as described in the 
preceding section is only one part of the description 
of the expected intensity distribution. There remains 
the problem of calculating the intensity in regions 
traversed by the rays. We already know that this in¬ 
tensity loss will not exactly obey the inverse square 
law except in very special cases. 


















52 


RAY ACOUSTICS 


We restrict ourselves to the case where the sound 
velocity is a function only of the depth coordinate y. 
The ray pattern in the xy plane can be computed 
according to the methods of Section 3.3. We can get 
the entire ray pattern in space by rotating the ray 
pattern of the xy plane about the vertical (y) axis; 
because the velocity depends only on y, the ray pat¬ 
terns in every plane through the y axis will be identi¬ 
cal in size and shape. 

We assume that the projector is a point source 
located on the y axis at the depth y 0 , which radiates 
energy at the rate of F energy units per unit solid 
angle per second. Then, energy will be projected into 



Figure 14. Specification of solid angle. 

a very small solid angle dil at the rate of Fdil energy 
units per second. The rays bounding this solid angle 
will curve in some fashion depending on n{y) and 
the angle of emission; suppose at the point P some¬ 
where out along the ray bundle, the cross-sectional 
area of the bundle is dS. Then the intensity at P will 
be the energy crossing this area dS per second, which 
equals Fdil, divided by the cross-sectional area dS. 

dil 

Intensity at P = F—• (49) 

dS 

Because of the cylindrical symmetry of the rays 
with respect to the y axis, we shall find it convenient 
to define our small solid angle as indicated in Figure 
14. It is the solid angle swept out in space by rotat¬ 
ing the portion of the xy plane between the angles 0 O 
and do -f dd about the y axis. On a unit sphere the 


solid angle so defined intercepts a spherical zone of 
radius cos do and width dd 0 which is therefore of 
area 2ir cos d 0 dd 0 . Thus our solid angle dil is given by 
dil = 2ir cos 6od9 0 . (50) 

We wish to calculate the intensity for the ray of 
initial direction d 0 in the xy plane, at the horizontal 
range R. By equations (49) and (50), this intensity is 
given by 

F■ (2tt cos doddo) 


I(R,d 0 ) = 


dS 


(51) 


where dS, the cross-sectional area, is clearly the area 
swept out when the segment PP' in Figure 15 is 
rotated about the y axis. We proceed to calculate dS. 



Figure 15. 
formulas. 


Diagram used in deriving intensity 


The horizontal range R is clearly a function of the 
depth h and d„: 

R = R(h,d o). (52) 

Therefore, the horizontal separation dR at a fixed 
depth is given by 


SR 

dR = - dd 0 . 

dd 0 


(53) 


The minus sign is inserted because R at fixed depth 
decreases as d 0 increases. 

Let d h be the direction of the ray at the point 
(R,h) as in Figure 15. Then PP', the shortest (normal) 
distance between these two rays near P is dR sin dh = 
— (dR/dd 0 )dd 0 sin d h . By rotating PP' about the y 
axis we get dS, the desired cross section of our bundle. 
This area is clearly that of a spherical zone with 
radius R and thickness— (dR/dd 0 ) sin d h dd 0 ; its area 
dS is therefore 

dR 

dS = —2ttR — sin d h dd 0 . 
dd 0 


( 54 ) 














CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 


53 


Substituting this expression into equation (51), we 
obtain for the intensity I at the range R the expres¬ 
sion 


I(RA) 

F 


COS 00 
~~dR. 

R — sin 0 A 
00o 


(55) 


It is now necessary to obtain expressions for R and 
for the partial derivative of R with respect to 0„. We 
begin by calculating the range R. We have 

h ^ r»h 

-j-dty = I cot Bdy, (56) 

v„ dy J y 0 

which, upon substituting cos 0 = (c/co) cos 0 O from 
equation (26), becomes 


R — cos 0 O 


J 

J U 


cdy 


yy Co - 


C 0 — C 2 COS 2 00 


(57) 


We differentiate this expression for R as a product of 
functions of 0 O , assuming that the usual formulas for 
differentiating under the integral sign are valid. When 
the two resulting integrals are put over one denomi¬ 
nator, the whole expression for the derivative simpli¬ 
fies to 


dR 

00o 



cdy _ 

(Co — c 2 cos 2 0o) 3 


(58) 


Substituting the expressions for R and for 0/2/00o, 
equations (57) and (58), into equation (55), we find 
for the intensity / the expression 



1 


(59) 

f" 

sin 0 O sin d h 1 

J » 0 i 

dy 

fin 0 

r h d v 

Jy 0 sin 3 0 

sin 0 = | 1 — 

(-J 

cos 2 0 O 

(60) 

sin e h = j/ 1 — 

©’ 

cos 2 0 O . 

(60a) 


For application, this formula suffers from two de¬ 
fects. First, it is not sufficiently simple; second, it is 
not sufficiently general. The second point will be 
taken up later, where it will be seen that equation (60) 
does not cover the important class of conditions 
where the sound ray becomes horizontal anywhere 
en route. As for the first point, we shall simplify 
equation (59) for application to these cases where it 
is valid. 

Under ordinary circumstances, c does not vary be¬ 
tween the sea surface and operational depths by 


more than 5 per cent of the surface velocity. As a 
result, those rays which leave the projector at a 
moderate angle will not become so steep that the 
sine of the angle cannot be replaced in good ap¬ 
proximation by the angle itself. Thus we may re¬ 
place the expression cl — c 2 cos 2 0 O , which appears 
three times in equation (59), by the approximation 

cl - <* cos 2 9„ « cs[l - - «S>] 

« CM - 2.), (61) 


in which e, as in Section 3.3.2, stands for the ex¬ 
pression (c — co)/cb. As a result, we can replace equa¬ 
tion (60) by the approximate relation 


7 

F 


/-— c h d v r h d y 

e ‘ v * - 


2e) i 


(62) 


while the range is given approximately by the expres¬ 
sion 


r' dy 

J v 0 V 0q — 2e 


(63) 


In most transmission work, the sound field inten¬ 
sity is reported either as transmission loss or as 
transmission anomaly. The transmission loss H is 
defined as the ratio of the source strength F and the 
sound field intensity in decibels, 



(64) 


The transmission anomaly A is defined as the ratio 
of the intensity predicted by the inverse square law 
and the sound field intensity I, also in decibels, 

F R- F 

A = 10 log — = 10 log— • (65) 

On the basis of this definition and equation (62) the 
transmission anomaly will be given by the approxi¬ 
mate relationship 



+ 10 log 0 O + 10 log B h , (66) 

where 6(y) = — 2e. 

Where it applies, this formula is simple enough to 
lead readily to results of practical significance, as 
under “Layer Effect” in Section 3.4.2. 

From its mode of derivation the expression (59) for 





















54 


RAY ACOUSTICS 


the intensity at a point P on a ray is not valid if at 
any place between the projector and P the ray has 
become horizontal. For, at a point where the ray is 
horizontal, 0 = 0, which implies that Co — c cos 9 0 
= 0, by equation (26). This means that the inte¬ 
gral in equation (57) becomes infinite at points where 
the ray is horizontal and cannot be differentiated 
under the integral sign. We therefore conclude that 
the expression (59) is valid only for rays that are 
always climbing or always dropping, but cannot be 
used, for example, to examine the intensity near 
places where the ray diagram predicts shadow zones. 

We now derive an expression for I(9 0 ,R ) which will 
be valid even at points on a ray beyond where the ray 
has become horizontal. This will be done by deriving 
an expression similar to equation (58) in which the 
variable of integration is 9 instead of y. 

In all cases, we have, because of equation (56), 


R = r cot 9dy 

J l/o 


Since 


dc 

Id 


it follows that R 


dy dc 

cot 9-)-d9. 
dc d9 


d I - cos 9 1 Co 

— Co- =-sin 9, 

d9L cos 0 O J cos 0 O 

C e * dy 

- I cos 9—d9. (67) 

hJ e„ dc 


Co 

COS 0o«. 


This expression for R has the advantage over equa¬ 
tion (56) in that the integrand does not become 
infinite for 0 = 0. This expression can, therefore, be 
differentiated with respect to 0 O even when the ray 
passes through points at which it is horizontal. 

The variable 0o occurs explicitly in the factor in 
front of the integral and as the lower limit of inte¬ 
gration and implicitly in the terms 9 h and dy/dc. 


Taking this into consideration, 

it follow 

s that 

dR 

Co sin 00 

r-dy 

1 — cos 9d9 

ho dc 



00o “ 

cos 2 0 O *- 




Co sin 0 O 
cos 3 0 O - 

r d-y „ , 

1 — cos- 9d9 

> 60 dc- 




Co sin 0 n r 

cos 2 dJ dy\ 

cos 2 0,,/ 

Ml 


cos 2 0 O L 

sin 9 h V dc)h 

sin 0 O V 

dc) o_ 


Though the expression (68) is much lengthier than 
the expression (58), it has the advantage of being 
valid at points on the ray where 0 = 0. The resulting 
intensity, calculated by using equation (55), will also 
be valid at all points on the ray. The quantities R and 
dR/d9 0 must be substituted from equations (67) and 


(68). These expressions can be simplified by means 
of the assumption that all angles are small. With this 
assumption all cosines of angles can be replaced by 
one; the sines may be replaced by the angles them¬ 
selves; and among a number of terms those multi¬ 
plied by higher powers of the angles may be disre¬ 
garded. The simplified expressions for R and for 
dR /00 o then take the form 



(69) 

(70) 


From equations (55) and (65) we have, as a general 
expression for the transmission anomaly *4, 


A = 10 log' 


'dR 

d9n 


sin 0;, 


R cos 0 O , 


(71) 


If we assume that 0 O and 6 >, are both so small that 
cos 0 O can be replaced by one, and sin 9 h by 9>,, 
formula (71) becomes 



(71a) 


By putting in the approximate values of R and 
dR d9 0 from equations (69) and (70), an explicit ex¬ 
pression can be obtained for the transmission 
anomaly. 

In the application of equations (67) to (70) one 
precaution must be taken. While the integrands of 
the integrals that occur remain finite when the angle 
of inclination becomes zero, these expressions ap¬ 
proach infinity as the gradient approaches zero. They 
are, therefore, not suitable for the treatment of 
propagation through isovelocity layers. 

Another method of computing the transmission 
anomaly that may be used whether or not a ray has 
become horizontal and is in a more convenient form 
for numerical computation is given under “Combina¬ 
tion of Linear Gradients” in Section 3.4.2. 


3.4.2 Applications 

Section 3.4.1 was devoted to deriving formulas for 
the intensity out along a ray as a function of the hori¬ 
zontal range and the velocity-depth variation. These 
formulas involve line integrals and are too compli¬ 
cated to use for practical intensity computations. 













CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 


55 


The formulas are simplified in this section by using 
various simplifying assumptions concerning the 
velocity-depth variation. 


Direct Beam in Linear Gradient 


Let us assume that the sound velocity increases or 
decreases linearly with depth, with the gradient a. 
Then, 


a - 


dc 

dy' 


(72) 


Since the velocity is never constant with increasing 
depth in this case, the approximate equations (09) 
and (70) are applicable. Using these equations, we 
find that the range R is given by the expression 


R = - ~(0 h - e 0 ) 

a 


(73) 


same linear velocity gradient assumed under “Direct 
Beam in Linear Gradient” in Section 3.4.2. First we 
shall assume one bottom reflection. This situation is 
pictured in Figure 16A where the ray hits the bottom 


R -— 



OCEAN BOTTOM 
A ONE REFLECTION 




OCEAN BOTTOM 
B SEVERAL REFLECTIONS 


and that the derivative of the range with respect to 


6 0 is given by 


dR = 1/ _ 0 O \ 

dd 0 a\ dj 


(74) 


Substituting these expressions into equation (71a), 
we obtain for the transmission anomaly the expres¬ 
sion 

.4 = 10 log 1 = 0. (75) 

The transmission anomaly vanishes, at least in this 
approximation. If we had used the rigorous expres¬ 
sions (G7) and (68) for R and dR/dd n , and the exact 
form (71) for the transmission anomaly, the following 
formula would have been obtained, which is rigor¬ 
ously correct, 

A = 20 log cos d 0 . (76) 

It may seem surprising that the transmission 
anomaly (76) does not depend on the sharpness of the 
velocity gradient. This seeming discrepancy results 
from the use of the horizontal range R in the defini¬ 
tion (65) of the transmission anomaly instead of the 
slant range r. 

The results (75) and (76) for this case of uniform 
downward refraction apply only to the sound field 
at points actually reached by the direct rays. If the 
water is very deep, there are portions of the ocean 
where no sound ray penetrates, as illustrated in 
Figure 24; in such regions the ray theory predicts a 
vanishing sound intensity and thus an infinite trans¬ 
mission anomaly. 


Reflected Beam in Linear Gradient 

We now calculate the intensity along a ray which 
has suffered one or more bottom reflections, for the 


Figure 16. Reflection of sound ray from sea bottom. 
A. One reflection. B. Several reflections. 


at an angle d b (d h > 0) and is reflected at the angle 
— d b . The rays will be refracted downward, as indi¬ 
cated; and the incident and reflected rays will be 
circular arcs with equal radii. 

We can compute the horizontal range R by equa¬ 
tion (69), which is valid for all cases where the ray 
path is made up of several arcs, provided care is 
taken in breaking up the interval of integration cor¬ 
rectly. 


R = — Co 


•J do ClC d, 


h dy 
et, dc 


dd 


- (d b — do) — — {dh + d b ) 

a a 


— — — (2 d b + d b — 0o). 
a 


To use equation (71a), we must also calculat e dR/dd 0 . 
Using equation (74), we obtain 

dR C°/ $o\ Co/^ — 

dd 0 ci\ d b ) a\ d b ) 

= —(2 — - + ~) • 

a\ d b dj 


Substitution of these expressions for R and dR/dd 0 
into equation (71a) gives 


A = 10 log 



2 d b + d b — do 


(77) 


For the case of. a ray suffering n + 1 reflections, 
pictured in Figure 16B, the procedure is similar ex- 












56 


RAY ACOUSTICS 


cept that n complete journeys from the bottom back 
to the bottom must be added to the interval of 
integration. The calculated transmission anomaly for 
a ray which leaves the source at the angle 0 O , suffers 
n + 1 bottom reflections, and strikes a receiving 
hydrophone at the inclination 0*, turns out to be 


A - 10 log 


00 0fc ~J 
2 (w + 1) 06 + 0 a — 00 


(78) 


Layer Effect 

When sound originates in an isovelocity layer or in 
a layer with a weak velocity gradient and then passes 
into a layer with a sharp negative gradient, the sharp 
refraction results in an extra spreading of the sound 
rays and a consequent drop in intensity. This phe¬ 
nomenon is called layer effect, and is of operational 
importance. We shall consider only rays which leave 
the projector in a downward direction, so that the 
formula (66) for the transmission anomaly will apply. 

Two separate cases will be treated. First, we shall 
consider the velocity-depth pattern shown in Figure 
17: an iso velocity layer, followed by a layer of 


VELOCITY. 



Figure 17. Bending of ray by temperature discon¬ 
tinuity. 


negligible thickness with a very sharp gradient and 
a total drop of sound velocity of amount Ac, followed 
in turn by a second isovelocity layer with the velocity 
Co — Ac. If the ray direction in the first isovelocity 
layer is 0 O , and in the second iso velocity layer 6 h , we 
have by Snell’s law (26) 

cos 6 h Co — Ac Ac 

cos 0 O Co Co 

If the angle 0 is small, we may replace its cosine by 
its approximate equivalent 1 — 0 2 /2. Using this ap¬ 
proximation, and dropping the negligible term 
(Ac/c o )0^, the preceding equation becomes 



If hi is the height of the sound source above the 
abrupt velocity change, and h 2 is the depth of the 
hydrophone below the velocity change, we easily find 
from formula (66) that 


A = —10 log 


+ 10 log 


(- + -) 

\0o ej 

+ 

mi UVi-l^YI 

- 10 '°imv + hx) J 


(ti + h\ 

\0o el) 


-f 10 log 0 O + 10 log 6 h 


(80) 


hi ht 

R * t + T • 


Next we shall consider the velocity-depth pattern 
shown in Figure 18: an isovelocity layer extending to 


VELOCITY 



Figure 18. Bending of ray by deep thermocline. 


a depth hi below the sound source, followed by a layer 
of indefinite extent with the constant velocity gradi¬ 
ent — a. At a depth y' below the top of the gradient 
layer the sound velocity will be c 0 + ay'. We there¬ 
fore obtain the following expression analogous to 
expression (79) for the ray direction d(y') at the 
depth y'. 

0(1 /') = A / 0q 2 (a/ Co)y f . (81) 

We shall use equation (66) to calculate the intensity 
of the sound received by a hydrophone at a depth h 2 
below the top of the gradient layer. Since the ray 
direction in the isovelocity layer is constant at 0 O , the 
separate integrals in equation (66) have the values 

= hi p dy' 

Jo 0 0o J o a/0q — 2(a/co)y' 

The last term may be integrated directly, and with 
use of equation (81) 

r h dy = hi + (00 - 9 k) 

Jo 0 0 O a/co 

= fh , h 2 

00 1(00 + 0 /,) ’ 































CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 


57 


VELOCITY SEA SURFACE 




Rn 


Figure 19. Ray path in succession of linear gradients. 


since d 2 0 — 0 2 h has the value 2(a/co)ho by equation (81). 
Similarly, we have 

+hj„. h f*h 2 


C h dj = hi f' 
Jo 0 s 0o + Jo 


dy' 


hi 


+ 


1 


(0o “ 2(a/ Co)y')' 

ho 


hi (0o — 0 a) 

0o (a/Co)0o0 A 


0o 0fl0A§(0O + 0*) 

Substituting these expressions into equation (66), we 
find for the transmission anomaly 

1 hi 

+ 


.4 = 10 log 


= 10 log 
since from equation (37), 
R 


|"0(A//ti 1 ho y 

L R V0^ + 0O0A i(0O + 0*)/- 
I - (hh i/ fh 0q \ 

L Repy + /ii§0 A (0 o + 0 a)/J’ 


(82) 




^(00 + 0ft) 

If the gradient is sharp, and if the range is con¬ 
siderable, the angle 0 O will generally be small com¬ 
pared with the angle at the hydrophone, B h . Also, if 
the hydrophone is not too far down, we may assume 
that the fraction h 2 /hi is not too large. In that case, 
the second term in the parentheses in both equations 
(80) and (82) is small compared with unity and may 
be omitted as negligible. In either case we have, then, 
as a rough estimate of layer effect, the simple rela¬ 
tionship 


A = 10 log 


(83) 


Combination of Linear Gradients 

In this subsection we shall derive a formula for the 
intensity along a ray which has passed through a suc¬ 
cession of layers in each of which the sound velocity 
changes linearly with depth in the layer. This con¬ 
dition is of considerable practical importance since 
most velocity-depth curves can be replaced in good 
approximation by a number of linear gradients. 

The assumed velocity-depth pattern is shown in 
Figure 19. There are n + 1 layers, labeled 0, 1,2, 3, 

• • • n, in which the velocity gradients are a 0 , ai, • • • a n 
respectively; the term a* represents the velocity 
change in the fth layer in velocity units per foot of 
depth increase in the layer labeled i, where i takes 
any integral value from 1 to n. The velocity at pro¬ 
jector depth is c 0 ; at the top of layer 1, Cijat the top of 
layer 2, c 2 ; and so on, as indicated in Figure 19. The 
ray direction is 0o at the projector, 0i at both the 
bottom of layer 0 and the top of layer 1, and so on; 
and, finally, 0„ at the bottom of the (n + 1) layer, 
which is assumed to be the depth of the receiving 
hydrophone. The total horizontal range covered by 
the ray is R; the component of horizontal range 
covered in each layer is designated by R 0 , R\, • • • R„, as 
indicated. 

We shall compute the intensity at the range R by 
means of the formula (71), which is generally ap¬ 
plicable. To use this formula, we must first derive 
an explicit expression for dR/ d0 () in terms of param- 


























58 


RAY ACOUSTICS 


eters which may be calculated from the given veloc¬ 
ity-depth distribution. 

Theraypath in the ith layer will be an arc of a circle 
whose radius is c,-/(a,cos 0 O ) according toequation (33). 
Let the small ray element ds be inclined at the angle 0, 
as in Figure 20. In traversing this small distance, the 



Figure 20. Ray path in ith layer. 


ray travels horizontally a distance ds cos 0. By equa¬ 
tion (32), we see that ds is given by the expression 
— Cid0/(a,iCOsdi); and the element of horizontal range 
covered by the ray in a distance ds is 

-—— cos Odd. 

a, cos 0; 


To get the horizontal range covered by the ray in its 
entire journey through the ith layer, we must inte¬ 
grate this result between 0, and 0, +1 . 


Ri = 


! 


— Ci 


e, di cos 


cos Odd 


--—(sin 0i — sin 0 1+ i) (84) 

di COS di 

Co sin di — sin 0, + i 

COS d 0 di 


because of Snell’s law (26). The results of this para¬ 
graph apply without changes of sign both to layers 
where the velocity increases with depth and to layers 
where the velocity decreases with depth. 

The total horizontal range R from the projector to 
the receiver is the sum of the range components in the 
?i+l layers. 


i=0 COS do i=0 


sin di — sin 

a, 


(85) 


where the symbol 2 indicates summation. By dif¬ 
ferentiating both sides of equation (85) with respect 
to do 


OR 

dd 0 


c 0 sin do s i n 0' — sin 0>+i 
cos 2 do i=o di 


+ 


C<> Y' 1 ( a ddi 

- 2 ^ —l cos di — 

cos do 1=0 ai\ ddo 


COS di + 1 


M<+i \ 
dd 0 / 


which may be written 
OR Co sin 0 ( 
ddo cos 2 do i 


Z-(sin0, - £ 
t=o aA 


+ 


COS di COS do dd; 


sin 0 l+ i 

COS 0;+ 1 COS 00 ddi + i 


sin d 0 


ddo 


sin 0o 


000 


)• 


( 86 ) 


For equation (86) to be usable, we must calculate 
dd i/ ddo. By Snell’s law, 


cos 0, 


Ci 

- cos 0 O . 

Co 


(87) 


By differentiating both sides of this with respect to d 0 
dd, d sin 0 O 

00 o Co sin 0, (88) 

sin 0 O cos 0, 
cos d 0 sin 0, 


By using dd, dd 0 from equation (88), and a corre¬ 
sponding expression for 00, + i/00 n , the expression in 
parentheses in equation (86) becomes 


sin 0, — sin 0,+i -f- 


cos 2 di cos 2 0,- 


sin di sin d i+ i 


1 

sin 


sin 0, 


Thus equation (86) become; 
OR 

ddo 


Co sin 0o yp 1 


cos'-0 O i=o a Asm 
sin 0i 


Asm di sm 0,+i/ 


z 


Ri 


cos 0 0 1=0 sin 0, sin 0, + i 

Putting this value of OR 00 o into formula (71), and 
noting that d h is simply 0 n+ i, we obtain the final result 

Ri 


A = 10 log 


sin 0 O sm 0 n+ , 


z 


R cos 2 0o i=osin0, sin 


(89) 


The expression (89) is in a form well-suited for 
practical intensity calculations. The various angles 0,- 
can be computed from the known velocity-depth pat¬ 
tern by equation (26), and the Ri can be obtained 
either from equation (84) or equation (36). 


Formulas for Transmission Anomalies 

In this section, the formulas obtained for the 
transmission anomaly resulting from refraction will 
be summarized. 

In the absence of a velocity gradient, or if the 
sound velocity increases or decreases linearly with 
depth below the projector, the transmission anomaly 
in the direct beam is negligible. 

If the sound velocity decreases linearly with depth 































RAY DIAGRAMS AND INTENSITY CONTOURS 


59 


below the projector, and if the ray has been reflected 
once by the ocean bottom, the transmission anomaly 
is given in good approximation by the equation 


.4 = 10 log 



*0 0 b \ 

e b + ej 


2 6b + Oh ~ On 


where 0 b is the angle of inclination at the bottom. In 
the case of multiple bottom reflections the transmis¬ 
sion anomaly is given by 


A 


10 log 


fe ( 2 (”+i) ~ i + 1) 

2 (n -(- 1 )0 b + 0), — 6 o 


where the number of bottom reflections is n + 1. 

The transmission anomaly for sound propagated 
through a thermocline (layer effect) is approximately 
given by 


.4 ~ 10 log 


/ h\6i\ 


where hi is the height of the projector above the top 
of the thermocline, 0 O is the inclination of the ray in 
the overlying isovelocity layer, and 0 h is the inclina¬ 
tion of the ray at the receiver. 

The transmission anomaly for sound propagated 
through a succession of layers, each of which possesses 
a constant gradient of velocity, is given by 

'"sin 0o sin 0 n +i Ri 


A 10 log 


(- 


£ 

R cos 2 0 O »=o sin 0, sin 0,+p 


where the various terms are defined under “Combi¬ 
nation of Linear Gradients” in Section 3.4.2. 


3.5 RAY DIAGRAMS ANI) INTENSITY 
CONTOURS 

3.5.1 Methods 

The differential equations (25) which govern the 
path of a ray in a medium where the sound velocity 
depends only on one coordinate cannot be easily 
integrated if the velocity depends on depth in a very 
complicated manner. We have seen, however, that 
the integration can be accomplished, and the path 
of a ray with a specified initial direction calculated if 
the depth interval is divided into layers in each of 
which the velocity gradient is constant. For this 
reason, rays are traced in practice by replacing the 
actual velocity-depth curve with a series of straight- 
line segments, as in Figure 8. 


The bending of sound rays in the ocean is too 
slight to be evident in a drawing that uses the same 
scale for range and depth. The deviation from 
straight-line propagation is never more than a few 
hundred feet in a mile; although such deviations are 
extremely important to a surface vessel seeking a 
submarine with echo-ranging gear, they do not show 
up well on paper unless the depth scale is expanded. 
For this reason, ray diagrams cannot be constructed 
geometrically in practice as the sum of circular arcs 
through the various layers. Instead, the change in 
range as the depth increases must be computed alge¬ 
braically as described under “Combination of Lin¬ 
ear Gradients” in Section 3.4.2, and the results 
plotted on a graph with suitably chosen scales. 

A special circular slide rule has been invented to 
simplify this calculation. 2 This instrument, developed 
by WHOI early in the war, gives the horizontal range 
covered by a ray in its passage through a layer with 
a constant gradient. It does this by using several 
scales arranged for convenience as concentric circles. 
The thickness of the layer, the temperature at the 
beginning and the end of the layer, and the direction 
of the ray at the projector are given to start with. By 
use of the slide rule one calculates directly the direc¬ 
tion of the ray when it enters and leaves the layer; 
from the average of the two directions the horizontal 
range covered in the layer can then be computed by 
use of equation (36). The instrument exactly dupli¬ 
cates the calculations described in Section 3.3 and 
avoids the necessity of consulting trigonometric 
tables. Since the direction of the ray as it enters the 
following layer is given as an intermediate step in 
calculating the range in the layer, the process may be 
duplicated until the ray has been traced through all 
the layers. This slide rule may also be used to com¬ 
pute intensities by integration along each ray since 
the scales provided may be used for evaluating equa¬ 
tion (90), which appears later. Similar mathematical 
aids were developed by UCDWR, and by other re¬ 
search groups doing a large amount of ray tracing. 3 

Another instrument developed by NDRC for the 
facilitation of ray tracing is the sonic ray plotter, 4 
pictured in Figure 21. The ray plotter is a device 
which integrates the differential equations (25) me¬ 
chanically and exhibits the solution not as an alge¬ 
braic function but as a curve denoting the ray path, 
drawn, of course, with a much expanded depth scale. 
The ray plotter has one advantage over the slide rule 
— it can plot the ray paths for any type of velocity- 
depth variation, no matter how complicated. 











60 


HAY ACOUSTICS 


The accurate plotting of many rays is facilitated by 
use of a method called the method of proportions 
(see reference 2). When this method is used, a few 
rays are drawn with the aid of a slide rule; the posi¬ 
tions of intervening rays can be estimated rapidly by 
an interpolating process. A full description of the 
method of proportions is given in reference 2. 

If the ray plotter or the method of proportions is 
used, it is not difficult to obtain a ray diagram with 
many rays drawn for closely adjacent values of 0 O , the 
ray inclination at the projector. From such a diagram 


Figure 21. The sonic ray plotter. 


the intensity at anv point can be determined graphi¬ 
cally by measuring the vertical separation between 
rays at that point. In most situations this is the 
simplest method for computing approximately the 
theoretical intensities in the sound field. 

The basic equation used in this graphical procedure 
for determining the sound intensity may readily be 
derived from the analysis in Section 3.4.1. By com¬ 
bining equations (49) and (50) the following results 
for the intensity / 

dd o 

I = 27 rF cos 0077 , ■ 
do 


The area dS is given by 

dS = 2vK cos ddh, 


where dh is the vertical distance between the two rays 
at the point where the intensity is measured, and R 
is the horizontal range. By combining these formulas, 
and by substituting into equation (65) for the trans¬ 
mission anomaly the following equation results 


.4 


= 10 log 



10 log R. 



For most cases of practical importance, cos 0 and 
cos 0o may be replaced by one; dh and dd may be re¬ 
placed by finite increments Ah and A0. Thus, we 
have, finally, the simple result 


A 



10 log R. 


(90) 


In equation (90), Ah and R must be expressed in 
yards; while A0 must be given in degrees. Although 
this equation usually gives sufficiently accurate re¬ 
sults, it is difficult to apply practically in regions 
where the intensity is changing rapidly, such as near 
the shadow boundary below an isothermal layer. 

The practical application of equation (90) is given 
in reference 2, which includes a graph giving the 
theoretical intensity I in terms of the measured ray 
separation in feet at the range R and the initial 
angular separation of the rays in degrees. 


3.5.2 Ray Diagrams for ^ arious 
Temperature-Depth Patterns 

In practice, the ray paths are usually computed 
not from the velocity-depth curve, but from the 
temperature-depth curve obtained with a bathy¬ 
thermograph. This is done because the sound velocity 
is very sensitive to changes in temperature of the 
magnitude usually encountered in the ocean and rela¬ 
tively insensitive to changes in pressure and salinity. 
The effect of pressure, although small, is usually al¬ 
lowed for in the drawing of rays because it is con¬ 
stant, causing an increase of 0.0182 ft per sec in 
sound velocity per foot increase of depth. The effect 
of salinity on the ray paths is usually ignored, except 
near regions where fresh water is continually mixing 
with ocean water; in such cases, the velocity-depth 
pattern must be calculated explicitly by use of both 
the bathythermograph record and the salinity-depth 
variation. 

The following paragraphs describe ray diagrams 
for various commonly observed temperature-depth 
patterns. A more detailed explanation of ray dia¬ 
grams along with explicit diagrams for some 380 
temperature-depth patterns of the sort found in the 
ocean is given in a report by WHO 1. 3 

Very Deep Isothermal Water 

In deep isothermal water all the rays show slight 
upward bending because of the constant effect of 
pressure. This bending, for a ray leaving the pro¬ 
jector in a horizontal direction, amounts to about 







RAY DIAGRAMS AND INTENSITY CONTOURS 


61 


50 ft in a distance of one mile. Figure 22 is the ray 
diagram for this ease. 

Isothermal Layer above Thermocline 

The most common temperature-depth distribution 
observed in the ocean possesses a surface layer of 
reasonably constant temperature, which overlies a 



Figure 22. Ray diagram for deep isothermal water. 


predicted low-intensity zone which lies between two 
zones of higher intensity has been verified in experi¬ 
ments with explosive sound. The experiments are 
described in Chapters 8 and 9. However, experiments 
with single-frequency sound, which are designed to 
test whether or not the beam splits when predicted, 
have frequently failed to indicate any splitting of the 
beam at all. The reasons for this discrepancy are not 
completely understood. Diffraction of sound into 
the low-intensity zone, although predicted to a 
limited extent by wave acoustics, is not sufficient to 
explain why the beam does not split. Possibly the 
sound in the predicted low-intensity zone may be 
largely due to scattering of sound either by in- 
homogeneities in the predicted path of the rays, or 
by roughness of the sea surface, or by irregularities 
of the temperature distribution in the ocean. 


layer where the temperature decreases rapidly with 
increasing depth, called a thermocline. A ray diagram 
for such a temperature-depth pattern, with the sur¬ 
face mixed layer extending down to a depth of 100 ft, 
and an underlying thermocline is shown in Figure 23. 
It will be noted that all the rays which issue from the 
projector at higher angles than 1.44 degrees remain 
entirely within the top layer; the rays become hori¬ 
zontal at some depth less than 100 ft and bend back 
to the surface. All the rays leaving the projector at 


TEMPERATURE-F RANGE IN YARDS 



thermocline. 

angles lower than 1.44 degrees reach the thermocline 
while still inclined downward; the thermocline pro¬ 
gressively increases this downward bending. I or 
theoretical reasons, then, the beam should split into 
two parts; one heads back toward the surface and the 
other heads down into the thermocline. Between these 
two beams the sound intensity should be very low 
according to the ray theory. 

All velocity-depth patterns for which the ray 
theory predicts such a splitting of the beam have 
been called split-beam patterns. The existence of the 


Strong Negative Gradient 
Negative temperature gradients are a frequent oc¬ 
currence near the sea surface, especially when the sur¬ 
face is receiving more heat than it is losing. Under 
such temperature conditions the rays are bent 
strongly downward. Figure 24 gives a ray diagram 

TEMPERATURE'? RANGE IN YARDS 



Figure 24. Ray diagram for strong negative tempera¬ 
ture gradient. 


drawn for a case where the negative temperature 
gradient amounted to about 8 F per 100 ft of depth. 
It is clear from the figure that the ray which left the 
projector horizontally has been bent down 400 ft by 
the time it has covered a horizontal range of 1,000 yd. 
The most important quality of the ray diagram for 
this case is the indicated shadow cast by the surface. 
It is clear from the figure that no ray leaving the pro¬ 
jector can possibly penetrate into the zone marked 
shadow zone if the water is deep. All rays lower than 
the ray leaving the projector at a climbing angle of 
4.8 degrees stay within that ray and are bent down¬ 
ward. The 4.8-degree ray itself becomes tangent to 
the surface and then bends downward. All rays higher 
than this “limiting ray” are reflected by the surface 


























62 


RAY ACOUSTICS 


TEMPERATURE-F 
0 |— 


100 


200 


300 



0 

100 

200 

300 


80 


75 


1000 


RANGE IN YARDS 
2000 


3000 


TEMPERATURE-F 
4000 80 




-65 DBl 

_——-— 


\jm . 

r£s 

£_ I 

3 ' 

n 

D 

CD \ 



in 1 
m | 

s \ 







\ 



50-FOOT ISOTHERMAL LAYER 
OVER SHARP THERMOCLINE 





/ 


/ 


57 


80 


77 



1000 


RANGE IN YARDS 
2000 


3000 


OVER SHARP THERMOCLINE 


150"FOOT ISOTHERMAL LAYER 
OVER WEAK THERMOCLINE 


OVER WEAK THERMOCLINE 

Figure 25. Sample intensity contour diagrams. 


FROM SURFACE DOWN 


4000 





inside the place where the limiting ray hits it; and so 
all the surface-reflected rays remain inside the 4.8- 
degree ray, also. 

If the actual sound intensity obeyed the predic¬ 
tions of the ray diagram, no sound at all would pene¬ 
trate out to horizontal ranges of more than a couple 
of thousand yards in the top 500 ft of the ocean. Thus, 
a submarine further from the projector than 2,000 yd 
could be almost certain of escaping detection by sonar 
gear. In practice, as with split-beam patterns, the 
shadow zone in the strict mathematical sense of a 
region of zero intensity does not exist. However, un¬ 
like split-beam patterns, the transmission anomaly 
invariably increases sharply at or near the indicated 
separation of sound from shadow when the down¬ 
ward refraction is strong. As discussed in Section 5.4, 
such zones of weak sound are observed whenever the 
temperature gradient is strong enough to cause the 
predicted shadow zone to start nearer the projector 
than about 1,000 yd; the sound in them is about 
30-40 db weaker than would be predicted by the in¬ 
verse square law. If the negative gradient is weak, 
however, so that the predicted shadow zone does not 
begin until a range of about 1,500 yd or more, sound 
usually decreases gradually and at a more uniform 
rate; the shadow zone in such a case can scarcely be 
said to have any real existence. Possible mechanisms 
for the penetration of sound into predicted shadow 
zones are discussed in Section 3.7. 


3.5.3 Intensity Contours 

Not all the characteristics of the sound field become 
apparent from a glance at the ray diagram. Although 
the distribution of intensity is governed by the 
spreading of the rays, the degree of spreading cannot 
be accurately estimated visually, and it is even diffi¬ 
cult to judge qualitatively. If a ray diagram is avail¬ 
able, the intensity at a point may be quickly esti¬ 
mated by measuring the vertical separation of the 
rays nearest that point, in accordance with equation 
(90). Since it is assumed that the ray bending is in a 
vertical direction only, the predicted sound intensity 
will be directly proportional to the measured separa¬ 
tion of the rays. 

Intensity contours provide a very graphic method 
for displaying the results of such computations. In 
practice, the intensity loss is usually reported in 
decibels below the sound level on the axis at a dis¬ 
tance of 1 yd from the projector. The exact value of 
the spreading loss at maximum echo range depends 
on many factors, such as the strength and direction¬ 
ality of the projector, the efficiency and operating 
condition of the gear, the intensity of background 
noise, and the amount of intensity loss due to absorp¬ 
tion and scattering. In many cases, it is useful to 
know at what range the intensity loss due to spread¬ 
ing will be 55 db, at what range it will be 60 db, etc. 
It is clear that the range at which the spreading 
loss has a specified value will depend on the depth of 


































































































VALIDITY OF RAY ACOUSTICS 


63 


the point to which the range is measured, and on re¬ 
fraction conditions, or more specifically on the tem¬ 
perature-depth variation indicated by the bathy¬ 
thermograph. 

The intensity contour diagram is a set of lines 
drawn on a ray diagram indicating the intensity loss. 
On each contour the intensity loss has a constant 
value, in a fashion similar to the curves of constant 
barometric pressure on a weather map. The contours 
are obtained from a ray diagram by using one of the 
methods discussed in Sections 3.4 and 3.5. On each 
ray, or for each pair of adjacent rays, the intensity, 
or transmission anomaly, is computed at suitably 
chosen intervals. Then one finds, by interpolation, 
the points where the intensity loss is 55 db, 60 db, 
65 db, and so on. After this process is carried through 
for all the rays, intensity contours can be drawn by 
joining the points of equal transmission loss on all the 
rays. 

Sample intensity contour diagrams for the oceano¬ 
graphic situations treated in Section 3.5.2 are given in 
Figure 25. The contour diagram for isothermal water 
is shown for comparison since it indicates optimum 
sound-ranging conditions, that is, the intensity losses 
which would be observed if the water had no tem¬ 
perature gradients, and if there were no attenuation 
losses; for this situation, the intensity loss out to the 
range R is given by the inverse square law and 
amounts to 20 log R. The contour diagrams for the 
split-beam cases are identical with that for the iso¬ 
thermal case at depths near the sea surface and at 
short to moderate ranges; at depths below the ther- 
mocline, however, the predicted spreading loss is 
much increased; the amount of increase depends on 
the depth to the thermocline and the sharpness of the 
thermocline gradient. In the case of downward refrac¬ 
tion, the intensity contours which denote large values 
of the intensity loss are piled together in the vicinity 
of the predicted shadow boundary. 

A more detailed discussion of intensity contours 
with a derivation of some of the basic equations de¬ 
rived at the beginning of this chapter is given in a 
report by UCDWR. 6 Sample theoretical intensity 
contours for different temperature patterns are also 
discussed in this reference. A comparison of these pre¬ 
dicted intensities with sound intensities found from 
explosive pulses is given in Chapter 9. The encyclo¬ 
pedia of ray diagrams in reference 5 includes intensity 
contours on most of the diagrams and thus may be 
used to find the type of predicted sound field for many 
different varieties of temperature-depth patterns. 


It will be seen in Chapter 5 that the intensity pre¬ 
dictions of the contour diagram are not, in general, 
sufficiently accurate to be trusted for the prediction of 
maximum echo ranges. However, they are useful for 
various special purposes, such as indicating howsound 
intensities should vary with depth at a fixed range. 

3.6 VALIDITY OF RAY ACOUSTICS 

In Sections 3.1 to 3.5 of this chapter the method of 
ray acoustics has been presented as an independent 
theory without much connection with the rigorous 
treatment of wave propagation presented in Chap¬ 
ter 2. We first noted in Section 3.1.1 that the im¬ 
portant features of the propagation of spherical waves 
could be derived equally well by using the concept of 
wave fronts connecting points which have equal 
phase of condensation, and the concept of energy 
transported by rays perpendicular to these w T ave 
fronts. Then w'e generalized the definition of wave 
fronts and rays, derived differential equations for the 
ray paths from these definitions, and solved these 
differential equations for the ray paths and the re¬ 
sulting sound intensity. 

It is important to remember, hov r ever, that the 
method of wave fronts for the general case placed no 
requirement on the wave front, except for stipulating 
that it be of the form (7) for some function W(x,y,z). 
To make the idea of wave fronts intuitively signifi¬ 
cant, it was implied that the wave front should always 
join points of constant phase of condensation;but this 
implication was never used. The ray paths depended 
only on the form of the function W and the variation 
of c; the intensity calculations depended, in addition, 
on the assumption that energy is transported out 
along the rays. In this section, where w r e try to find 
a connection between ray acoustics and w T ave acous¬ 
tics, w r e must assume a physical significance for the 
wave fronts. Accordingly, we shall make the explicit 
assumption that the w r ave fronts join points of equal 
phase of condensation since we already know that 
the assumption brings ray acoustics and wave acous¬ 
tics into agreement for the case of spherical w'aves. 

In this section, we shall examine whether w r ave 
acoustics and ray acoustics with this definition of 
w'ave fronts are equivalent in general or only under 
some special conditions. Since sound field calcula¬ 
tions are much simpler by the ray method than by a 
rigorous solution of the wave equation, it will be ex¬ 
tremely valuable to know' when the ray theory can 
be applied without much error and w'hen it will lead 
to definitely w'rong results. 




64 


RAY ACOUSTICS 


3.6.1 Eikonal W ave Fronts versus 
General Wave Fronts 

It will be remembered that the entire method of 
rays was based on the eikonal equation (13), which 
in turn was based on the assumption that the wave 
fronts (7) “grow” perpendicularly to themselves. 
That is, the eikonal equation was derived by assum¬ 
ing that the wave front at time t + dt is found from 
the wave front at time t by moving each point on the 
latter a distance cdt along the outward normal. We 
shall now show that wave fronts ordinarily do not 
obey this law of propagation rigorously, but that the 
assumption often provides a good approximation. 

It is intuitively apparent that wave fronts, defined 
purely as surfaces of constant phase without refer¬ 
ence to the way they grow, exist in the exact case, at 
least when the dependence on time is harmonic. We 
shall define these wave fronts in the rigorous case by 

V(x,y,z ) = Co(t - t 0 ) (91) 

reserving the expressions IF for those cases where the 
wave fronts grow perpendicularly to themselves, and 
where IF therefore satisfies the eikonal equation. We 
shall call surfaces (91) general wave fronts, and sur¬ 
faces defined by similar equations, with F replaced 
by IF, eikonal wave f ronts. 

We know that in instances where the sound source 
vibrates harmonically with a single frequency / the 
solution of the wave equation can be expressed as the 
real part of the complex expression 

p = A(xpy,z)e 2Hf{t - LV{x ' y ' z)Vc ^. (92) 

This expression is identical with equation (6), except 
that we assume that the expression (92) with the 
function V(x,y,z) is a rigorous solution of the wave 
equation, while the expression (6) with the function 
W(x,y,z) was obtained by means of a Huyghens con¬ 
struction so that W(x,y,z) would satisfy the eikonal 
equation. 

We now shall see under what conditions the ex¬ 
pression (92) can satisfy the wave equation and, 
simultaneously, V(x,y,z ) satisfy the eikonal equation. 
Suppose p satisfies equation (27) of Chapter 2, and 
simultaneously F satisfies the eikonal equation (13). 
The latter condition is 

© + © + © “ n! " 0 (93) 
The former condition may be simply calculated by 
noting that equation (92) may be written as 

p = giog A-Znifi.VMfrif.t' (94) 


Substitution of the expression (94) into the wave 
equation, performance of the indicated differentia¬ 
tions, and collection of terms is a straightforward 
calculation which will not be reproduced here. The 
real and imaginary parts must vanish separately; 
these parts are 


© 


+ 


+ 


, X 0 jd 2 (Iogd) 

— n~ — — \ -——■ 

4 tt t dx 2 


, d 2 (logd) , d 2 (log.4) , 
“t r© r „ 0 ~r 


+ 


and 

d-V 


dy- 
d(log A) 

- dy - 

d 2 ! 
dz‘‘ 

d V <9(log A) 


+ 


dz 2 
d(log A) 
. dz 


d(logd) 

dx 

*\ 


= 0. (95) 


d 2 I 

-(- --f---|- 2 

dx- dy- dz 2 


d V d(log .4) 
_dx dx 
, dV d(\og 




dy dy ' dz dz 
Clearly, F will satisfy condition (93) only if 
©d 2 (log.4) , d 2 (logyl) , d 2 (log .4) 


(96) 


+ 


d(log .4) 
dx 


+ 


' a (log A) 

- dy J 


+ 


This can happen if X 0 is zero, or if 


'd(log A) = 0 
(97) 


dz 


i/d 2 A d°-A a-.4 \ 
A \ dx 2 dy 2 dz 2 / 


= 0, 


(98) 


since the expression in braces in (97) easily reduces 
to the above. This condition (98) is usually not 
satisfied. While it happens to be satisfied by the pres¬ 
sure wave of a point source in a homogeneous 
medium, it does not hold, for instance, for the radia¬ 
tion of a double source. In general, equations (93) 
and (95) will be rigorously equivalent only if the 
wavelength X 0 vanishes. 


3.6.2 Conditions for Nearly 
Eikonal Wave Fronts 

We derived in Section 3.6.1 the conditions under 
which wave fronts, defined as expanding surfaces of 
constant phase of condensation, expand perpendicu¬ 
larly to themselves. It is more useful to know how 
large the frequency must be, relative to the other 
parameters of the problem, before the function 
V(x,y,z ) of equation (92) very nearly satisfies the 
eikonal equation; we will then know under what con¬ 
ditions the wave fronts are very nearly perpendicu¬ 
larly expanding. 






























SHADOW ZONE AND DIFFRACTION 


65 


Clearly the expression B of equation (98), the re¬ 
mainder term will be negligible compared with the 
other terms if 

X 0 (log A)' « V (99) 

Ao(log A)" « (F') 2 , (100) 

where the prime denotes any spatial derivative, and 
« means “is negligible compared with.” If F even 
approximately satisfies the eikonal equation (13), 
then 

V' ~ n, (101) 

where the symbol ~ signifies “is of the same order of 
magnitude as.” 

Another useful relation is obtained from equation 
(96). The functions A and F must satisfy equation 
(96) as long as the surface (91) has the significance 
of a general wave front. But equation (96) implies 
that 

F"~F'(log AY, (102) 

which in turn implies that 

AoF" ~ F'X 0 (log A)' « V'V' (103) 

because of equation (99). Combining equations (103) 

and (101), 

X 0 V" « n 2 . (104) 

In the ocean the index of refraction n is of the order 
of magnitude of unity. Then, the relation (104) may 
be stated in the following words. The first spatial de¬ 
rivative of F must not change much over a spatial dis¬ 
tance of one wavelength. The first spatial derivatives 
of F give the direction of the rays; while the second 
derivatives, yielding the rate of change of ray direc¬ 
tion, give the curvature of the rays. Therefore, the 
condition (104) becomes the following. The direction 
of the ray must not change much over a distance of 
one wavelength. In regions where the ray curves 
very strongly, ray acoustics cannot be applied safely. 

Differentiating the eikonal equation (13), we get 
V'V" ~ nn' or V" ~ n' because of equation (101). 
In view of equation (104), this means that 

A 0 n' « w 2 ~ 1. (105) 

In other words, the index of refraction must not 
change much over a distance of one wavelength. 

We derive one more restriction — this time on the 
amplitude function A. From equations (102) and 
(104), we also have 

X 0 (log A)' < 1. (106) 

The relation (106) means that log A must not change 
much over a distance of one wavelength. Since this 
change is very nearly Ao A'/A, this means that the 


percentage change in A over one wavelength must be 
very small. 

We can summarize our conclusions as follows. The 
eikonal equation usually will not lead to a good ap¬ 
proximation (1) if the radius of curvature of the rays 
is anywhere of the order of, or smaller than, one wave¬ 
length, or (2) if the velocity of sound changes ap¬ 
preciably over the distance of one wavelength, or (3) if 
the percentage change in the amplitude function A is 
not small over the distance of one wavelength. 

3.6.3 Comparison of Ray Intensities 
and Rigorous Intensities 

It follows from the results of Section 2.7.3 that if 
the general wave fronts are defined by equation (91), 
and the instantaneous acoustic pressure by equation 
(92), then the rigorous intensity is given by 



and, further, that the direction of energy flow is char¬ 
acterized by the direction numbers dV/dx : dV/dy : 
dV/dz. The latter direction is perpendicular to the 
general wave front; thus, if the wave fronts are eikonal 
wave fronts, the energy flows along the rays in the 
rigorous case. If the wave fronts are approximately 
eikonal wave fronts, then the directions perpendicular 
to these wave fronts represent very nearly the true 
direction of energy flow. 

Thus, if the conditions for eikonal wave fronts de¬ 
rived in Section 3.6.2 are satisfied, the energy ema¬ 
nating from the source into all solid angles will re¬ 
main within the tubular confines assumed in deriving 
the ray intensity. We can therefore say, intuitively, 
that if the wave fronts are very nearly eikonal wave 
fronts, the ray intensity will be very close to the 
rigorous intensity. Further, we can say that in both 
cases the intensity will be given by 



3.7 SHADOW ZONE AND DIFFRACTION 

When the velocity decreases from the surface down¬ 
wards, the ray theory predicts a sharp shadow 
boundary across which no sound ray penetrates; a 
typical ray diagram for such an instance is shown in 
Figure 24. At the shadow boundary the ray theory 






66 


RAY ACOUSTICS 


predicts a discontinuous drop of intensity from a 
finite value on one side to a zero value on the other. 
It was shown in Section 3.6 that the ray theory can¬ 
not be trusted whenever it predicts such a rapid 
change of intensity in a distance of only a few wave¬ 
lengths. Thus, it is necessary to use the wave equa¬ 
tion directly to compute the intensity of sound which 
penetrates the so-called shadow zone. 

The simplest case of a shadow zone is that pro¬ 
duced by a screen in front of a light source. As shown 
in Figure 26, the ray theory predicts that no light 

SOURCE 




// // i -nA 

' v \\ 

/ '/ / \\ s 

/ /. SHADOW ZONE \. ' 

/ // / ' ' 
/ / / / 


/ 


/ 


V \ 


/ / 


\ \ 

\ \ 
\ ' 


Figure 26. Optical shadow zone produced by screen. 


can reach the shadow zone behind the screen. When 
the rays carrying the energy are curved, as in Figure 
24, it is the surface of the ocean that intercepts the 
curved rays and “casts a shadow.” In either case, 
however, some energy actually appears inside the 
predicted shadow zone, and the wave is said to be 
“diffracted.” 

The computation of diffracted sound in the shadow 
zone is a rather complicated problem in the general 
case. To indicate the type of analysis required, and to 
show the general nature of the results, a simplified 
problem will be considered here. As shown in Figure 
27, a sound projector is assumed to be placed against 
a vertical wall, which extends down to great depths. 
The introduction of the wall simplifies the problem 
without changing the final results essentially. The 
water is assumed to be so deep that bottom-reflected 
sound may be neglected. The projector face is as¬ 
sumed to be so wide that the horizontal spreading of 
the sound beam may be neglected; thus, only the 
two-dimensional problems need be considered. The 
sound velocity c is assumed to vary according to the 
law 


where B is a constant, and y represents depth below 
the surface. Since B is in practice very small, this 
gradient is indistinguishable from a linear gradient 
at depths of interest. The exact velocity gradient at 
the depth y is given by 


dc _ cl B 
dy 2c (1 + By) 2 


( 110 ) 


Thus, at the surface, where y = 0, the velocity 
gradient — b is given by 


Bco dc 

2 dy y=o 


( 111 ) 


The gradient (109) is chosen instead of a simple 
linear gradient not for physical reasons, but because 
it simplifies the following computations. 


tPROJECTOR SURFACE X —•> VELOCITY-^ 



Figure 27. Sound shadow cast by sea surface. 


To solve the wave equation under these conditions, 
it is necessary to use the method of normal modes 
developed in Chapter 2. In particular, we must find a 
solution to the wave equation (27) in Chapter 2 which 
satisfies the boundary conditions we shall impose. As 
in Section 2.7.2, we look for a solution which is the 
product of three functions, one dependent only on 
the time t, another dependent on the depth y, and the 
third, a function of the horizontal distance x. The 
coordinate z need not be considered in the two- 
dimensional case under discussion. 

Following the analysis of Section 2.7.2, we there¬ 
fore write 

p(x,y,z,t) = F(y)G(x). (112) 

By substitution of equation (112) into the wave equa¬ 
tion (27) of Chapter 2, and by dividing through by 
C 2 , it is found that F and G satisfy an equation of the 
form 


(PF JPG 4ir 2 / 2 

G— + F— + -~-FG = 0. 
dy- dx~ c 2 


c- 


1 + By 


(109) 


(113) 













SHADOW ZONE AND DIFFRACTION 


67 


If equation (113) is divided through by FG, and equa¬ 
tion (109) used for c, 



~l drF 
-Fdy 2 + 


dTT -p 
c 2 o 


(1 + By) 


= 0. 


( 11 - 1 ) 


Since the first bracket depends only on x and the 
second only on y, equation (115) can be satisfied only 
if each bracket is constant. If we denote the first 
bracket by — y-, the second bracket must be +y 2 , and 
we have 


(FF 

dy- 


+ 


■hrp 
L Co 


(1 + By) - y 1 


iPG 

T7 + y~G = 0. 
(Ix- 


F = 0 (115) 

(116) 


The basic problem is to find solutions of equations 
(115) and (116) which satisfy the boundary condi¬ 
tions. First, we have the boundary conditions for 
equation (115). In the analysis in Section 2.7.2, these 
boundary conditions were that the pressure vanished 
both at the surface and at the bottom. Here, also, 
the pressure must vanish at the surface. However, 
the water is so deep that the condition at the bottom 
disappears. Instead, there is simply the condition 
that at some distance below the projector no sound 
is coming upwards; that is, any sound present at these 
depths is coming down from shallower depths. Al¬ 
though this boundary condition is somewhat compli¬ 
cated to formulate exactly, the general result is the 
same as that found in the solution of equation (161) 
of Chapter 2. In this earlier instance it was found 
that sin 2iry/\, n corresponding to F(y) in equation 
(112), when B is zero, satisfied the two boundary con¬ 
ditions only if \ u had one of a number of fixed values. 
Similarly, the function F(y) can satisfy the two bound¬ 
ary conditions only if y has one of a certain number 
of values. These values, which are called characteris¬ 
tic values of y, may be denoted by yi, yo, y s , and so on, 
or more generally by y h where,/ can be any integral 
number. For each of the characteristic values yj, 
equation (115) has a particular solution F j(y) which 
satisfies the boundary conditions. 

Once a value of yj has been chosen, the solution of 
equation (116) is very simple. For each value of yj, 


G = A ,e~ wx 


(117) 


where Aj is an arbitrary constant. a Thus the wave 


a The negative sign must be taken in the exponent so that 
Pi in equation (118) will correspond to a wave moving away 
from the projector; that is, p, must be a function of 2irft — y,x, 
where yj is positive. 


equation is satisfied by any product of the type 

Vi = A fi vin F y (y)e- l V. (118) 

Equation (118) satisfies the boundary conditions at 
the surface and at great depth since Fj(y) satisfies 
these conditions. However, the boundary conditions 
at the vertical plane x = 0, the assumed vertical 
wall, must also be satisfied. These conditions are that 
the particle velocity at the sound projector must be 
v 0 cos 2-irft, and that the particle velocity at all other 
points in the plane x = 0 must be zero. 

To satisfy this boundary condition at the plane 
x = 0 requires a combination of an infinite number of 
possible solutions of the form (118). Each A, must 
be chosen in such a way that the sum has the re¬ 
quired properties. Methods for doing this have been 
developed, but are beyond the scope of this discus¬ 
sion. However, the final result is that the pressure p 
is the sum of many terms of the type (118) with 
e- ,r ' ft the only common factor. 

Within the direct sound field a large number of 
these terms are important, and an exact computation 
is necessary to find p. In the shadow zone, on the 
other hand, one term dominates, and the other terms 
may be neglected. This is because all the yj are partly 
real, partly imaginary, with the result that the abso¬ 
lute value of exp ( iyjx ) decreases exponentially for 
sufficiently great values of x. It can be shown that 
the range at which only one term dominates is ap¬ 
proximately the range to the shadow boundary com¬ 
puted from the ray theory. This dominant term is the 
one for which yj has the smallest imaginary part. 
Thus, the theory predicts that in the shadow zone the 
sound intensity falls off exponentially with increasing 
range, or, in other words, that the predicted trans¬ 
mission anomaly in the shadow zone increases line¬ 
arly with increasing range. 

Although the exact determination of the different 
characteristic values y is somewhat involved, it is 
relatively simple to show how these values depend on 
the frequency/, the velocity gradient, and the sound 
velocity Co at the surface. This is useful since it indi¬ 
cates how the attenuation into the shadow zone may 
be expected to vary under different conditions. In 
order to investigate this dependence of yj on the 
other variables, we rewrite equation (115) in a simpli¬ 
fied dimensionless form. Let 


and 


4ir 2 / 2 


c 0 


(119) 


(120) 








68 


RAY ACOUSTICS 


where D is an arbitrary constant to be determined 
later. Then equation (115) becomes, on dividing 
through by D' 1 , 




4 TT 2 f 2 Bu \ = 
c 2 0 D 3 ) 


0. 


( 121 ) 


If D is chosen so that 


D 3 = 


iir-f-B 


Co 

then equation (115) becomes 


+ (K + u)F — 0 (122) 

du- 


where 


K = 


nc 0 


(x 2 / 2 5co) a 


(123) 


Equation (122) has solutions of the type desired only 
for certain characteristic values of K, denoted by the 
symbol K The different values of A', are determined 
only by the nature of the differential equation (122) 
and by the two boundary conditions, namely that the 
sound pressure is zero at the surface and that no 
sound is coming up from below the projector. Thus 
the values of A, are independent of the frequency, 
sound velocity, and velocity gradient. 

Once these characteristic values of K have been 
found, the corresponding values of y to be used in 
equations (115) and (116) can be found directly. By 
substitution in equations (123) and (119), we find 


2 47T 2 / 2 (ir-pBcnY 

Mi = - o -5-A 


(124) 


C 0 c 0 

The second term in equation (124) is always very 
much less than the first in cases of practical im¬ 
portance. Even for a temperature gradient as large 
as 1 F per ft of depth increase, and for a frequency 
of only 100 c, the second term is less than 1 per cent 
of the first for Ah less than 10, the region of practical 
interest. Thus we may take the square root of equa¬ 
tion (124), expand in a series, and retain only the 
first two terms. This process gives 


2tt 

X 


Bco VI 

8tt// J 


_ 2tt Ki/irfB *Y 

X ~ 4 V Co / 


Let A i be the characteristic value of A with the 
smallest imaginary part, and let this imaginary part 
be denoted by iK[. Let the theoretical sound pressure 
associated with the characteristic value Ah be pi. In 
the shadow zone the intensity is proportional to the 
square of V\ since the sound pressures associated with 
the other characteristic values Kj may be neglected. 


The intensity level found from equation (119) 


L = 20 log pi = C — 20 (log w e)- 


K\(nf IV¬ 


YS 

(126) 


where C includes Hi and the other variables taken 
over from equation (118). While C changes gradually 
with position, it is nearly constant along the shadow 
boundary. Multiplying out terms in equation (126), 
and using equation (lit) for B, we get, finally, 
5.05Ai/ s (— dc/dyYx 


L = C 


(127) 


It should be emphasized that equations (126) and 
(127) apply only in the shadow zone. In the main 
beam other terms corresponding to other values of 
A, must be considered. 

The analysis in a report by Columbia University 
Division of War Research 7 considers the radiation in 
three dimensions sent out by a point source and is 
thus more general than the simple analysis presented 
here. However, the final result for the sound in the 
shadow zone is nearly identical with equation (127); 
the only difference is that the term 5.05Aj becomes 
25.7 in the exact computation of reference 7. With 
this substitution, we have the following formula for a, 
the attenuation coefficient beyond the shadow bound¬ 
ary in decibels per unit distance. 

a _ 25.7 fH-dc/dy) . (12g) 


In this equation/is the sound frequency in cycles per 
second, and dc/dy is the velocity gradient in feet per 
second per foot. If Co is in feet per second, formula 
(128) gives the attenuation in decibels per foot; if c 0 is 
in yards per second, the result is the attenuation in 
decibels per yard. 

Since inverse-square spreading is quite negligible 
compared to the intensity drop at the shadow 
boundary, equation (128) gives the slope of the 
transmission anomaly at points beyond the shadow 
boundary. However, this equation cannot be used 
at shorter ranges and must therefore be regarded as 
an expression for the local attenuation coefficient in 
the shadow zone. 

Equation (128) is compared with observational 
data under Attenuation Coefficient at Shadow Bound¬ 
ary in Section 5.4.1, where it is shown that, the ob¬ 
served local attenuation coefficients beyond the 
shadow boundary are not more than about half the 
predicted values. In other words, in practice much 
more sound appears in the shadow zone than is pre¬ 
dicted by equation (128). 










Chapter 4 


EXPERIMENTAL PROCEDURES 


T he preceding chapter was concerned chiefly with 
the development of the ray-tracing technique, the 
earliest theoretical approach which led to practical 
results in the prediction of maximum ranges. This 
method was, however, only partially successful. Its 
chief accomplishment was the prediction of the 
shadow zone boundary in the presence of pronounced 
negative gradients at the surface. 

Predicted maximum echo ranges computed by ray- 
tracing methods agreed with the available observed 
range data to a fair degree of accuracy, but it was 
clear that these prediction methods w r ere too simple. 
The evidence relating maximum observed ranges to 
temperature conditions was too incomplete to be 
analyzed with a view' to improving range-prediction 
methods. Navy vessels could not often be made avail¬ 
able for range determinations under carefully con¬ 
trolled conditions, and the scattered observations 
made in the course of routine operations w^ere incon¬ 
clusive. It w'as decided, therefore, to initiate a pro¬ 
gram in which the sound field produced with standard 
Navy echo-ranging gear would be measured in much 
greater detail than before. It was contemplated that 
this study would place the prediction of sound ranges 
on a firmer basis and in general would lead to a better 
understanding of the basic factors important in trans¬ 
mission of sound through the ocean. Subsequently, 
this program was broadened to include sound of fre¬ 
quencies between 100 and 60,000 c, and to cover 
situations somewhat different from those encountered 
in routine operation of standard gear. Only such a 
broad experimental investigation of the propagation 
of sound under various conditions can possibly fur¬ 
nish an adequate insight into the mechanisms deter¬ 
mining the sound field in the sea. 

This chapter deals with the experimental methods 
which have been developed in connection with the 
sound field program. The results obtained will be 
discussed in Chapters 5 and 6. 


4.1 QUANTITIES CHARACTERIZING 
TRANSMISSION 

Before launching into a detailed discussion of these 
experimental methods, it will be necessary to review 
briefly the principal quantities which characterize the 
transmission of sound energy in the sea. In general, 
sound power is transmitted at a particular frequency 
or in a specified frequency band; all statements in this 
section concerning power, intensity, and sound level 
refer to the frequency or frequency band once speci¬ 
fied. 

Let F denote the power output per unit solid angle 
on the axis of symmetry of the sound source; at a 
moderate distance r from the source, the sound in¬ 
tensity on the axis therefore equals F/r 2 . The pow'er 
output per unit solid angle in any other direction wdll 
be given by bF, where b, the pattern f unction defined 
in Section 2.4.4, is a function of the direction; by 
definition, b equals unity for the direction of the 
projector axis. 

Since decibels are commonly used in sound field 
measurements, w r e shall transform F into a more con¬ 
venient quantity, the source level S. At a point on the 
axis at a distance of 1 yd from a point source, the 
sound intensity I n will be proportional to F. The 
source level S is defined as this sound intensity at 1 yd 
in decibels above a suitably chosen reference in¬ 
tensity /,■: 

£ = 10 l°gf^)' (!) 

The reference intensity I r is usually chosen as that 
corresponding to an rms pressure of 1 dyne per sq cm. 

Actual sound sources, such as a battleship gen¬ 
erating propeller and machinery noise, frequently 
have large spatial extensions, and the sound level 1 yd 
from the source is not well defined. However, at dis¬ 
tances large compared with the linear dimensions of 


69 


70 


EXPERIMENTAL PROCEDURES 


0 




150 200 


VELOCITY ANOMALY 
IN FT PER SEC 



Figure 1 . Transmission loss ( H) and transmission anomaly {A). 


such a source, the sound field intensity drops off like 
that of a point source producing similar sounds; in 
other words, for intensity calculations at long ranges, 
the actual source may be replaced by an equivalent 
point source. The reported source level of an ex¬ 
tended source is nothing but the sound level of the 
equivalent point source at a range of 1 yd. For echo¬ 
ranging transducers, the sound level is often meas¬ 
ured at a distance of a few yards and then extra¬ 
polated to a distance of 1 yd by means of the inverse 
square law. 

Consider now the sound field intensity 1 at some 
specified location, presumably at a fair distance from 
the sound source. This intensity is commonly ex¬ 
pressed as a sound pressure level L in decibels above 
an rms pressure of 1 dyne per sq cm (decibels of a 
pressure level are defined as twenty times the loga¬ 
rithm of the ratio between the rms acoustic pressure 
and 1 dyne per sq cm). If the sound source is highly 
directional, like an echo-ranging projector, it is 
usually understood that the projector is trained, that 
is, rotated about its vertical axis, toward the point 
at which the transmission loss is to be determined. 
But in the absence of a tilting device, the axis ray 


leaves the projector in a horizontal direction and may 
then be refracted to a depth different from that of the 
recording hydrophone. The difference S — L will 
therefore depend, in general, on the directivity pat¬ 
tern of the transmitter. As long as the distribution of 
acoustic pressure does not approach the conditions 
of explosive sound, S — L will be independent of the 
absolute power output of the transducer. The dif¬ 
ference S — L in decibels is called the transmission 
loss and is denoted by the symbol H. 

Frequently, the transmission loss is represented in 
terms of its deviation from the law of inverse square 
spreading. If this law vere valid, the transmission 
loss should amount to 20 log R, where R is the hori¬ 
zontal range from the transmitter to the chosen 
point. The expression H — 20 log R is called the 
transmission anomaly and is denoted by A. Figure 1 
shows the experimentally determined values of H and 
A in a particular run with plots of the sound velocity 
against depth and of the computed ray diagrams. 

H and A are quantities depending on the trans¬ 
mission characteristics of the path under considera¬ 
tion and on the directivity pattern of the transmitter. 
They are independent of the power output of the 


































TRANSMISSION RUNS 


71 



GOOD COHERENCE POOR COHERENCE 

Figure 2. Examples of good and poor coherence. 


transmitter; moreover, *4 is an absolute quantity 
which is independent of the system of units chosen. a 

Transmission loss and transmission anomaly are 
the principal quantities which characterize the propa¬ 
gation of sound from the source to any point of in¬ 
terest. For some purposes, it is also desired to obtain 
information on the steadiness of the transmitted sig¬ 
nal and on its “coherence.” Slow changes in signal 
strength that occur in the course of several minutes 
are called variation. Changes that take place in the 
course of seconds are called fluctuation. The coherence 
of a signal may be loosely defined as the degree of 
fidelity with which the envelope of the transmitted 
signal is duplicated by the envelope of the received 
signal. If transmission conditions in the ocean did not 
change rapidly, one would be a perfect copy ot the 
other, except for a negligible transient. Actually, 
conditions sometimes change so rapidly that the 
shapes of the transmitted and received signal re¬ 
semble each other only slightly. Figure 2 shows (A) a 
case of good coherence, and (B) a case of poor coher¬ 
ence. Both of these figures show oscillograms of re¬ 
ceived supersonic signals recorded on the same equip¬ 
ment. 

A detailed discussion of variation, fluctuation, and 
coherence will be given in Chapter 7. 

4.2 DETERMINATION OF TRANSMISSION 

LOSS 

Information on transmission loss has been ob¬ 
tained by three distinct methods: first, transmission 
runs; second, echo-ranging runs; and third, the sta- 

» A is ten times the logarithm of the ratio between the 
power flow per unit solid angle close to the source and the 
power flow per unit solid angle at the specified location, both 
of these quantities should be expressed in the same units. 
The apex of the solid angle is in both cases formed by the 
sound source. 


tistical analysis of observed echo and listening ranges. 
Sound transmission runs include all investigations in 
which the source of sound is separate from the receiv¬ 
ing instrument and in which the sound travels from 
source to receiver without suffering reflection from a 
target; slanting reflection from the surface or the 
bottom of the sea is, however, not excluded. While 
various setups have been used for transmission runs, 
the most common one involves the use of two ships. 
One ship carries the sound source, whereas the other 
ship is equipped with hydrophones whose outputs are 
recorded. In echo-ranging runs, the same transducer 
is used as both source and receiver. The sound is 
propagated to a target and then reflected back toward 
the point of origin. The target may be a vessel, but is 
more frequently an artificial target, that is, a device 
used exclusively for research and training purposes. 
The observed range information has been furnished 
to the research groups in the form of log books and 
patrol reports by naval craft on active duty. 

Of the three methods of investigation mentioned, 
transmission runs have proved by far the most power¬ 
ful and reliable tool. The other two methods, analysis 
of observed ranges and echo runs, are now merely 
subsidiary. 

4,3 TRANSMISSION RUNS 

The characteristic feature of the transmission run 
is the employment of separate devices for transmit¬ 
ting and receiving the sound. It is, therefore, possible 
to measure the transmission over any type of path by 
varying the depth of the projector, the depth of the 
hydrophone, and the horizontal distance between 
them. Depending on the instrumentation, it is further 
possible to vary other important acoustic parameters, 
such as signal frequency and signal length, or to em¬ 
ploy signals composed of several frequencies or a 
continuous range of frequencies. The temperature 









72 


EXPERIMENTAL PROCEDURES 


distribution in the ocean during transmission, the 
depth and nature of the sea bottom, and in many in¬ 
stances factors such as condition of the sea surface, 
wind velocity, the presence of ocean currents, and the 
presence of salinity gradients, will affect the trans¬ 
mission characteristics of the ocean; these must be 
recorded along with the geometry of the transmission 
path itself. In recent experiments, not only the level 
but also the coherence and the degree of fluctuation 
of the received signal have been studied. All in all, the 
number of variables determining a signal is almost 
overwhelming; also, the characteristics of the result¬ 
ing signal are quite complex. In any given investiga¬ 
tion, both field procedure and the analysis of data are 
necessarily concerned with only part of the complete 
picture. 

Ordinarily, the sound source in transmission runs 
is a transmitter driven by a harmonic oscillator 
through suitable amplifying stages, so that single¬ 
frequency sound is put into the water. Frequencies 
used range from 200 c up to 100 kc and more, but 
more runs have been carried out at 24 kc than at any 
other frequency. A second ship carries the receiving 
gear, hydrophone, amplifiers, and recorders. The 
hydrophones are usually cable-mounted hydrophones, 
which can be lowered to any desired depth from a few 
feet to several hundred feet below the surface. 

In the most common form of run, the depth of the 
hydrophone or hydrophones is kept constant during 
one run. The range, however, is varied during the run 
from 100 or 200 yd to several thousand yd, by having 
the sending ship either approach or recede from the 
receiving vessel. The run is usually completed in less 
than half an hour. It is hoped no major changes 
in temperature distribution or other oceanographic 
variables will have taken place during that time. A 
more detailed description of the field procedure will 
be given in Section 4.3.2. First, however, a brief 
description of sound-transmitting and sound-receiv¬ 
ing equipment will be given. 


4 .3.1 Sound Sources and Receivers 

A sound source suitable for transmission runs 
should satisfy several requirements. It should be 
easily controlled. It should be capable of being 
mounted on a ship or towed by a ship. Its output 
should be stable. Its frequency characteristics should 
be simple, that is, it should produce either single¬ 
frequency sound or wide-band noise with a smooth 


spectrum; and the acoustic power output should be 
high so that even at long ranges the received signal 
will usually be above the background. In practice, 
most results have been achieved with the use of single¬ 
frequency sources, such as echo-ranging projectors. 
Some work has also been done with noise makers of 
the type used for acoustic mine sweeping. 

Single-frequency sources have been of three kinds, 
electromagnetic or dynamic speakers for sonic fre¬ 
quencies, and magnetostrictive and piezoelectric pro¬ 
jectors for supersonic frequencies. Work has been re¬ 
ported by UCDWR at 200, 600, and 1,800 c, and 14, 
16, 20, 24, 40, 45, and 60 kc, and by WHOI at 12 and 
24 kc. More transmission runs have been carried out 
at 24 kc than at all the other frequencies combined 
because echo-ranging gear used by the Navy was de¬ 
signed for use at approximately that frequency. Oc¬ 
casionally, transmission runs have been made with 
“chirp” signals; these are frequency-modulated sig¬ 
nals in which the frequency rises linearly from 23.5 
to 24.5 kc or some similar frequency range during 
a pulse. 

Other important parameters of the sound source 
are its directivity and its power output. The direc¬ 
tivity may be reported in the form of pattern plots 
in the horizontal plane and in the vertical plane. Ten 
times the logarithm of b, the pattern function of the pro¬ 
jector, is plotted on a circular graph against the angle 
from the axis. These plots are incomplete since no in¬ 
formation is given concerning the value of b off the 
two planes plotted. Most echo-ranging projectors, 
however, approach rotational symmetry with respect 
to the axis so that a single plot including the axis 
gives adequate information on the pattern in all 
directions. Figure 3 shows the directivity pattern of 
the JK crystal projector which has been used by 
UCDWR for many transmission runs at 24 kc. 

Frequently, the directivity of a projector is re¬ 
ported by means of a single quantity, the directivity 
index D. The directivity index is defined (see Section 
2.4.4) by means of the equation 



in which ft denotes the full solid angle. The units are 
decibels. The directivity index so defined has the 
value of zero decibel for a spherically symmetric 
sound source. Since the axis for echo ranging is invar¬ 
iably the direction of greatest power output, b no¬ 
where exceeds unity, and D is a negative quantity. 
For the standard Navy sound heads QC (magneto- 



TRANSMISSION RUNS 


73 




Figure 3. Directivity patterns of the JK SK4926 at 24 kc. 


strietive) and JK (X-cut Rochelle salt), the direc¬ 
tivity index is approximately —23 db at 24 kc. 

The total power output of a projector is usually of 
less interest than the power output per unit solid 
angle on the axis. This quantity is customarily re¬ 
ported in terms of the source level S, which has al¬ 
ready been defined in Section 4.1. The source level of 
the gear used in the UCDWR transmission studies is 
about 107 db above 1 dyne per sq cm 1 yd from the 
projector face. 

The receiving instruments in transmission runs are 
usually cable hydrophones. The sound head consists 
of the electroacoustical element itself and sometimes 
contains a preamplifier which boosts the output 
voltage before it passes through the cable to the main 
sound stack. The electroacoustical element itself 
may be either a crystal element (as in the CN8 series 
used for a long time at UCDWR), or it may be a 
magnetostrictive device (similar to the Harvard 
Underwater Sound Laboratory [HUSL] B19-H). 

The receiving response of a hydrophone is defined 
as the ratio between the rms voltage across the out¬ 
put terminals of the hydrophone or the preamplifier 
and the sound pressure of a plane wave incident on 
the axis of the hydrophone. It is ordinarily reported 
in decibels above 1 volt per unit sound pressure and 
then denoted by s. Waves incident in directions not 


270' 



180° 

Figure 4. Response pattern of the CN-8-2 No. 597 
hydrophone at 24 kc in horizontal plane. 


parallel to the axis will produce lower voltages than 
sound waves of the same amplitude incident on the 
axis of the hydrophone; in other words, most hydro¬ 
phones discriminate against off-axis sound inputs. 







































74 


EXPERIMENTAL PROCEDURES 


0 



180 ° 


Figure 5. Response pattern of the CN-8-2 No. 597 
hydrophone at 60 kc in horizontal plane. 

The degree of discrimination is reported in a manner 
analogous to the statement of the directivity pattern 
of a projector. The ratio of sensitivity in a given 
direction to the sensitivity on the axis is denoted by 
b', which is often plotted in decibels relative to axis 
sensitivity. The degree of discrimination of the 
hydrophone may also lie reported as a single quantity, 
its directivity index, defined by the equation 



which is completely analogous to equation (2). 

Cable hydrophones cannot be trained since they 
are freely suspended from their cables. It is, there¬ 
fore, extremely important for a cable hydrophone to 
be nondirectional in the horizontal plane; otherwise 
appreciable unknown errors in the received intensity 
result. Several models, which have been used in re¬ 
search, come fairly close to nondirectionality. Figure 
4 shows the horizontal directivity pattern of the 
CN-8 crystal hydrophone, used extensively for trans¬ 
mission runs at both UCDWR and WHOI. This pat¬ 
tern was determined at a frequency of 24 kc. Figure 5 
shows the horizontal directivity pattern of the same 
hydrophone at (SO kc. It will be noted that at this fre¬ 
quency the CN-8 is quite noticeably directional in 
the horizontal plane. More recently the HUSL 



Figure 6. Horizontal directivity pattern of the Har¬ 
vard B19-H hydrophone at 20, 60, and 100 kc. 


B19-H magnetostrictive hydrophone has found 
favor because of its great stability and high degree of 
nondirectionality in the horizontal plane at a wide 
range of frequencies. Figure 6 shows the horizontal 
directivity of the B19-H at 20 kc, at 60 kc, and at 
100 kc. 

The response of a hydrophone is a measure of the 
strength of the electrical signal which will be passed 
into the cable with a given intensity of incident sound. 
Since the cable and subsequent amplifying stages will 
produce a certain amount of instrumental back¬ 
ground noise, the response alone may, under exceed¬ 
ingly favorable external conditions, determine the 
level of the minimum detectable signal. The thermal 
noise in a 1-c band is determined by the receiving 
response and by the effective resistance G of the 
hydrophone according to the following equation. 1 

N = 10 log G — s — 195. (4) 

G is measured in ohms while .V represents decibels 
above 1 dyne per sq cm. 

The output of the hydrophone, or preamplifier, is 
transmitted through the hydrophone cable into the 
receiving sound stack. There the signal is filtered, 
amplified, possibly rectified or heterodyned, and then 
fed into the recorder. Most commonly used for re¬ 
cording are cathode-ray or galvanometer oscillo¬ 
graphs with a very nearly linear' 5 response, which re- 

b The circuit is said to respond linearly if the amplitude of 
the recorded signal is proportional to the amplitude of the 
incident sound field. 






























TRANSMISSION RUNS 


75 



Figure 7. Oscillograph record of received 50-msec single-frequency signals; R is the radio signal, T is the 60-c timing 
trace, I and III are rectified traces, and II is the heterodyned trace. The range is approximately 80 yd. 


cord the received signal on film or sensitized paper 
moving past the oscillograph at a constant speed. A 
timing trace, usually provided, permits accurate 
measurements of time intervals on the film or paper 
strip. 

The signal is transmitted not only as an under¬ 
water sound signal, but also as an airborne radio 
signal over a radio link, usually FM, between the two 
ships. At the ranges involved, the radio signal arrives 
practically without time delay and without distortion. 
In addition to providing a convenient monitoring de¬ 
vice, the radio signal serves as a means of accurately 
determining the range between the two ships. Since 
it is reproduced as a separate trace on the oscillograph 
record, it is an easy matter, with the help of the tim¬ 
ing trace, to measure the time interval between the 
arrivals of the radio signal and the sound signal, and 
thus determine the distance traveled by the sound for 
each separate transmitted pulse. The resulting record 
will be similar to the strip in Figure 7. The top trace 
is the radio trace, the bottom trace the timing trace, 
and the three traces labeled I, II, and III, are the 
outputs of three different hydrophones, recorded 
simultaneously. The outputs of I and III were recti¬ 
fied before being recorded, and the output of II was 
heterodyned down to 800 c before recording, but not 
rectified. The range can be read off the record with 
an error of less than 15 yd. In the example shown in 
Figure 7 the range is approximately 80 yd. 

In the past, transmission runs have also been re¬ 
corded by means of power level recorders. These re¬ 
corders are electromechanical recording instruments 
with a logarithmic response. A stylus records on a 
moving paper strip the received signal level in 
decibels above the reference level. Although these 
records are much easier to read than oscillograph rec¬ 
ords, they suffer from a certain unreliability of the 
instrument. Frequently, the stylus “sticks”; that is, 
it follows a change in signal level only when this 


change exceeds an appreciable threshold value. Fur¬ 
thermore, the stylus travels only a certain number of 
decibels per sec (50 to 500 db per sec, depending on 
the model); the instrument, therefore, cannot record 
correctly the level of very short signals. For this 
reason, power level recorders have not been used in 
recent transmission work. 

WHOI has developed an electronic device designed 
to combine the advantages of both oscillograph and 
power level recorder. It consists essentially of a 
rectifier, an amplifier with logarithmic response over 
a range of approximately 80 db, and a galvanometer 
oscillograph. This device has a time constant of 
roughly0.5 msec. 2 Up to the present, it has been used 
only for reverberation studies; whether it will prove 
useful in transmission work remains to be seen. The 
amplifier used in this device has been improved since 
reference 2 was published. 

The output of the hydrophone is passed through 
filters at some stage before it reaches the recording 
instrument. The purpose of the filter is to improve 
the signal-to-background ratio. All the unwanted 
background (see Division 6, Volume 7) contains 
energy in a very broad frequency band. A band-pass 
filter centered at the signal frequency will discrimi¬ 
nate against the broad-band background in favor of 
sound at the signal frequency. Most of the filters 
used are approximately Y 2 kc wide. Such a width 
leaves an adequate margin for possible drift of the 
driving oscillator in the sending stack and for dop- 
pler. 

Both the amplifiers and the recording instruments 
will be linear and otherwise satisfactory only in a 
limited range of signal amplitude. On the other hand, 
actual signal levels are likely to change by as much 
as 80 db between short and long ranges of transmis¬ 
sion. For this reason, step attenuators are provided. 
These attenuators are usually operated by hand; 
however, in the most recent installation at UCDWR 











76 


EXPERIMENTAL PROCEDURES 


the attenuator is automatically actuated when the 
received signal level rises above or drops below certain 
limits for several successive signals. All changes in 
attenuator setting are recorded, either in a separate 
log book, or automatically on the oscillograph record. 3 

4.3.2 Field Procedures 4 

In this section the field procedures used in trans¬ 
mission runs will be described. First, a number of 
oceanographic facts are ascertained and recorded, 
either immediately preceding or immediately follow¬ 
ing each transmission run. These include the depth 
of the ocean, the type of bottom (in shallow water), 
the state of the sea, the swell, the wind strength, and, 
most important, the vertical temperature distribu¬ 
tion in the ocean. The bathythermograph, an instru¬ 
ment which measures vertical temperature gradients, 
is in general use in the Navy wherever echo ranging 
is involved. It is a recording device which can be 
lowered into the water down to considerable depth 
(as much as 450 ft for the “deep” model) and which, 
upon being returned to shipboard, indicates the 
temperature versus depth distribution as a trace 
marked on a smoked slide. Ordinarily, a bathyther¬ 
mograph is lowered on each of the two vessels partici¬ 
pating in a transmission run; the source vessel makes 
its lowering at the point of greatest distance from the 
receiving vessel and frequently one or two lowerings 
at intermediate points. Figure 8 shows a blank which 
contains the oceanographic information belonging to 
a simple transmission run. This blank has been used 
at UCDWR. 

Another subsidiary step is the calibration of equip¬ 
ment. In this chapter the term calibration will be 
used with a definite meaning. Calibration is a pro¬ 
cedure which translates sound field data taken off the 
oscillograph trace into the transmission loss. The 
transmission loss was defined in Section 4.1 as the 
difference in decibels between the source level S of the 
projector and the sound level L at the hydrophone. 
Since the source level of the projector is defined in 
turn as the sound pressure level at a range of 1 yd, 
the transmission loss is then the difference in decibels 
between the sound levels at a range of 1 yd, and the 
range r of the hydrophone. In theory, then, one would 
obtain the transmission loss according to the follow¬ 
ing formula. 

H = 10 log (^j , 


where a is the sound pressure amplitude in the water 
at the hydrophone, and d\ is the sound pressure 
amplitude at 1 yd. 

If it were possible to bring the receiving hydro¬ 
phone up to a distance of 1 yd from the projector, the 
absolute transmission loss could thus be readily de¬ 
termined without knowing either the projector source 
strength or the hydrophone response. The squared 
ratio between the signal amplitude (on the oscillo¬ 
gram or on the tube screen) at 1 yd and the signal 
amplitude at R yards would give the transmission 
loss, provided the design of the receiving stack guar¬ 
antees proportionality between received pressure 
amplitude and recorded trace amplitude. Actually, 
it is next to impossible to bring the projector of the 
sending vessel and the hydrophone of the receiving 
vessel closer together than about 30 to 50 yd without 
inviting a maritime catastrophe. Correction of the 
observed signal level at 30 or 50 yd back to the pre¬ 
sumed level at 1 yd has at times been done by 
straightforward application of the inverse square 
law. However, this method is probably too simple. 
There is some evidence that, even at ranges of 50 yd, 
the transmission loss cannot always be expected to 
follow the inverse square law of spreading. 4 

Several more complicated methods have been em¬ 
ployed, which in theory should enable determination 
of the absolute transmission loss. Although none of 
these suggested calibration procedures have proved 
completely satisfactory, some may be preferable to 
the simple correction by means of the inverse square 
law. The following paragraphs are devoted to a 
description of some of these more refined calibration 
procedures. 

During a substantial part of its supersonic trans¬ 
mission program, UCDWR carried out runs called 
calibration runs at very short range, approximately 
100 yd. During these runs, both the sending vessel 
and the receiving vessel were permitted to drift. The 
signal level at 100 yd, obtained from this run, was 
arbitrarily assigned a transmission anomaly value 
of zero; and all other transmission data obtained on 
the same day were referred to the 100-yd level 
obtained in the calibration run. Somewhat later, 
these special runs were discontinued. Instead, an 
average was taken of all the short-range data ac¬ 
cumulated during the day, and a value of zero for 
the transmission anomaly was assigned to this 
average. In these two methods no attempt is made 
to calibrate in terms of a distance of the order of 
1 yd; that is, no test is made which would relate 




VESSEL: REG- SEND _ WATER DEPTH _ LOCALII Y_ RUN_ 

BT NUMBER_ _ BOTTOM TYPE_ REG PCS. LAT_ LONG._ 

SLIDE Mo-- SEA_SWELL_ DATE—_ TIME!_ 

TIME OF BT- - WIND_M/H.WEA._ REEI-PROJECT No. 

PATTERN_ 


TRANSMISSION RUNS 


l i 


DEPTH IN FEET 



Figure 8. Blank for oceanographic information (UCDWR). 


























































































































































































































































































































































































































78 


EXPERIMENTAL PROCEDURES 


the sound level at 100 yd to the source level as 
defined in Section 4.1. 

WHOI used a related method of calibration until 
the summer of 1945. For each individual run, the ob¬ 
served sound field levels were each increased by 
20 log R and plotted against range. The resulting 
points between a range of 100 yd and the range where 
the observed intensity was 40 db less than the in¬ 
tensity at 100 yd were then fitted by inspection with 
a straight line. This line was extrapolated to a 
range of 1 yd to give the presumable sound level (in 
decibels above 1 gv) at that range. 

More recently, both institutions have put into use 
new methods, which are designed to obtain a calibra¬ 
tion directly in terms of the short-range (l to 10 yd) 
sound level. These methods are of two kinds, which 
may be described as unaided calibrations and as 
calibrations w ith the help of standards. As an example 
of unaided calibration, WHOI floats one of the re¬ 
ceiving hydrophones out to a distance where it can be 
picked up safely by the crew of the sending ship. The 
hydrophone remains connected by cable with the re¬ 
ceiving ship, and its output is recorded by the same 
equipment used in the transmission runs. The hydro¬ 
phone is then secured at a measured distance of a few 
yards from the face of the projector. The projector is 
trained on the hydrophone, and a signal is put into 
the water and received by the hydrophone. At this 
short range, the effect of the surface is minimized by 
the directivity of the projector, and tests have shown 
that the sound field obeys the inverse square law 
within the limits of observational accuracy. This 
method of calibration would be expected to yield the 
most accurate and most trustworthy results. A sub¬ 
stantially identical method is occasionally employed 
by UCDWR for checking the results of other calibra¬ 
tion methods. The unaided methods are not always 
practical in the field since in a heavy sea the transfer 
of a hydrophone from one ship to the other may not 
be possible. At best, the hydrophone transfer is a 
cumbersome and time-consuming maneuver. Never¬ 
theless, unaided calibration is standard procedure at 
WHOI. 

Most of the field calibrations at UCDWR are now 
made with the help of calibration standards, sound 
units which are designed primarily for this purpose 
and which are more stable than other units. Typical is 
a UCDWR procedure which involves the use of two 
OAX transducers; these transducers are HUSL de¬ 
signs. One of these two transducers is kept aboard 
the receiving vessel, the other aboard the sending 


ship. Aboard the sending ship, where the projector 
source strength S is to be determined, the OAX is 
used as a hydrophone. It is attached to a boom which 
can be swung over the side and which insures that the 
OAX is always at the same distance (13 ft) from the 
face of the keel-mounted JIv projector. The projector 
is trained on the OAX, and the voltage generated 
across the terminals of the OAX unit by the sound 
field of the JK projector is measured. Aboard the 
receiving ship, where the hydrophone response S is 
sought, the second OAX unit is hung over the side 
amidships to a depth where it clears the keel. The 
receiving hydrophone is also hung over the side, on 
the other side of the ship, and is lowered to the same 
depth as the OAX. The distance between the two 
units is, therefore, with fair accuracy, the beam width 
of the receiving ship. The OAX is then energized as 
a projector with a standard power input, and the gen¬ 
erated voltage across the terminals of the receiving 
hydrophone measured. If these two tests lead to the 
same results or very nearly the same results day after 
day, it is assumed that all four units are constant. 
Large jumps (several decibels) are presumably indi¬ 
cations that either one of the four sound heads or the 
electrical follow-up (amplifiers and associated equip¬ 
ment) has changed its characteristics. If the change 
cannot be assigned to the electrical follow-up, it is 
assumed that the standard OAX units have remained 
unchanged and that either the JK power output or 
the receiving hydrophone response has changed. In a 
word, the characteristics of the projector and hydro¬ 
phone used in the transmission runs are measured in 
terms of auxiliary standards, which are presumed to 
be stable. The standards themselves are thoroughly 
tested every few months at a special calibration 
station. 

For a time, WHOI also used a similar method of 
calibration that depended upon the use of HUSL 
monitor standards. This method of calibration was 
later abandoned by that group in favor of unaided 
calibration. 

Clearly, not one of the calibration methods which 
have been described is both convenient and wholly 
satisfactory as a method for translating observed 
hydrophone voltages into accurate estimates of the 
absolute transmission loss. Yet, until electroacoustical 
equipment is developed which can be relied on to re¬ 
main stable, field calibration remains a necessity. It 
is to be hoped that rapid and adequate procedures of 
calibration will be developed in the future. 

Once the equipment is calibrated, the transmission 



TRANSMISSION RUNS 


79 


loss can be determined by measuring the received 
signal level at the range and depth of the hydro¬ 
phone. There are several types of runs. It is possible, 
for instance, to make a vertical transmission run, in 
which the range between the two ships is kept very 
nearly constant and in which the hydrophone is 
slowly raised or lowered, so that the transmission loss 
is determined as a function of depth at a fixed range. 
In horizontal runs, the depth of the hydrophone is 
kept fixed, while the range is changed. Horizontal 



runs are much easier to carry out than vertical runs. 
In a vertical transmission run, the depth of the 
hydrophone can be changed only slightly each time. 
One or several pings are transmitted while the hydro¬ 
phone is kept at a constant depth; and then the 
hydrophone depth is changed again. Also, whenever 
the hydrophone is moving through water, the flow of 
the water past the hydrophone gives rise to noise, 
which may effectively mask the signal. In a horizontal 
run, the receiving ship is permitted to drift, or, in 
shallow water, anchored. The noise due to water 
current thus is minimized and the hydrophone cable 
tends to hang straight down. Proper training and 
control of depth is thus facilitated. The sending ves¬ 
sel then runs either toward or away from the receiv¬ 


ing ship. In this manner the range can be varied con¬ 
tinuously by an amount of several thousand yards 
without ever interrupting the transmission of signals. 
A supersonic transmission run of the horizontal type 
takes, on the average, about 20 or 30 minutes. If it is 
desired to obtain the transmission loss at several 
depths, two or three hydrophones can be suspended 
at various depths from the receiving vessel. The over¬ 
whelming majority of transmission runs made up to 
the present have been horizontal runs. 

In all transmission runs, elaborate precautions 
have always been taken to keep hydrophones at their 
nominal depth. Because of the wind drift of the re¬ 
ceiving vessel, and because of ocean currents going 
in different directions at different depths, deep 
hydrophones, which are lowered occasionally as far as 
450 ft below the surface, will rise to a much shallower 
depth unless special care is taken to make them hang 
straight. To this end, a 300-lb weight is suspended 
from a strong steel cable; the hydrophone hangs 
down from this weight and is held down by an addi¬ 
tional 25-lb weight as shown in Figure 9. The hydro¬ 
phone cable carries relatively little weight in this 
arrangement. 

Horizontal transmission runs can be either ap¬ 
proaching runs or receding runs, that is, the sending 
ship can either close or open the range. In the reced¬ 
ing run, the wake of the sending vessel is located be¬ 
tween the two ships. Since it has been found that 
wakes are capable of absorbing sound, the sending 
ship usually changes its course from time to time in a 



Figure 10. Course followed during a receding run. 


manner illustrated in Figure 10 in order that the 
direct sound path between the two ships may never 
pass through the wake laid by the sending ship. How¬ 
ever, over shallow bottoms where change of course 
would result in a changing bottom type, this pro¬ 
cedure is sometimes not followed. During an approach 
rim, the sending ship remains between its wake and 
the receiving ship, and it may, therefore, run along 
a straight course and pass the receiving vessel at a 
























80 


EXPERIMENTAL PROCEDURES 


range of approximately 100 yd. During a run, the 
sending vessel keeps its projector trained at all times 
on the receiving vessel. This aiming is done by means 
of a pelorus, mounted either on the flying bridge or 
vertically above the sound projector to eliminate 
parallax. In a recently developed installation, selsyn 
repeaters cause the sound projector to follow auto¬ 
matically the changes in bearing of the pelorus. In 
former installations, an operator in the ward room of 
the sending ship had to match the two bearings by 
hand. At WHOI, projector training is now completely 
automatic, with the help of a radio compass. 

In transmission work at supersonic frequencies, 
ping lengths are usually either about 50 msec or 
about 10 msec. It is believed that as the signal length 
decreases below 50 msec aural perception of the re¬ 
sulting echoes becomes more and more unsatisfactory 
(see Volume 9 of Division 6). Very short pings have 
an important use, however, in transmission studies. 
If the water is fairly shallow and the ping length is 
sufficiently short, the directly transmitted and the 
bottom-reflected signals can be examined separately. 
This is possible when the time resolution of the fol¬ 
low-up circuit is sufficient to resolve the time differ¬ 
ence of arrival between direct signal and bottom-re¬ 
flected signal. The minimum requirements of resolu¬ 
tion in a specific case will depend on the geometry of 
the paths, depth, range, and refraction pattern of the 
ocean. 

For very long ranges, the signals often arrive with 
badly distorted envelopes and with tails known as 
forward reverberation. When such tails are present, no 
resolution of the electrical circuit will result in satis¬ 
factory separation of the two sound paths. In the 
absence of such tails, resolution is frequently possible 
even with fairly long, square-topped signals; in other 
words, it is possible to distinguish three portions of 
the received signal trace, the direct signal alone, the 
composite signal, and the bottom-reflected signal 
alone. 

Signals are emitted at a rate of about one signal per 
second. Once a minute pinging is interrupted, and a 
long signal of 10 seconds’ duration is sent out. These 
long signals serve two purposes. First, a received long 
signal often provides a very instructive, graphic il¬ 
lustration of the degree of coherence of the transmis¬ 
sion. Furthermore, this long signal makes it possible 
to correlate the received short sound signals with the 
radio signal, and thus to determine the range at which 
the signal was received. In a transmission run carried 
out at 5,000 yd, by the time a sound signal arrives at 


the receiving ship, three additional signals have al¬ 
ready been put into the water. The once-a-minute 
breaks facilitate the identification of particular sig¬ 
nals. 

The overall accuracy of the determination of the 
transmission loss of an individual signal has been im¬ 
proved steadily in the course of the transmission pro¬ 
gram. A distinction must be made between the deter¬ 
mination of the absolute transmission loss, and the 
determination of the difference in transmission loss 
between two signals received at the same range, or 
one signal received at different ranges. The deter¬ 
mination of relative loss is not affected by errors of 
calibration, while the determination of the absolute 
loss is affected by calibration errors. The uncertainty 
of calibration in the earlier data taken both by 
UCDWR and WHOI is very large and probably ex¬ 
ceeds 10 db in many instances. Improvement in pro¬ 
cedure has cut this uncertainty down to approxi¬ 
mately 1 db. Both absolute and relative determina¬ 
tions are affected by training errors of the projector 
and by the horizontal directivity of the hydrophone. 
Training errors are small at long range where the 
bearing is changing slowly. At ranges of the order of 
100 yd, where the bearing changes rapidly, training 
errors can be significant even when great care is used 
in following the target. The uncertainty of training 
has been almost eliminated by improved instrumenta¬ 
tion and is probably well within 1 db at the present 
time. Even in earlier work, training errors probably 
never caused an error in received sound level much in 
excess of 1 db. The most recent hydrophone models 
in use are practically nondirectional at 24 kc, but the 
directivity of the CN-8 model used in earlier studies 
introduced errors of about 2 db. Thus, the experi¬ 
mental error of most of the transmission loss deter¬ 
minations at UCDWR is probably about 2 db, while 
for the most recent data the experimental error is 
probably considerably smaller. 

4 .3.3 Analysis of Data 

This section will be concerned with the analysis of 
data in which single-frequency supersonic sound is 
received by one of the recording systems with a linear 
response. The procedure used in the analysis of runs 
carried out with sonic sound will also be sketched. 

Figure 11 shows that the received sound intensity 
is subject to rapid changes in intensity, which obvi¬ 
ously cannot be related to observed changes in range 
or temperature distribution. Figure 12 shows the re¬ 
ceived amplitude of a continuous, 10-sec, 24-kc signal. 



TRANSMISSION RUNS 


81 


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Figure 12. Two 10-soc signal oscillograms. 


































































































82 


EXPERIMENTAL PROCEDURES 


The upper strip was obtained at a range of 110 yd in 
the direct sound field, while the lower strip is a typical 
record of sound received at a long range, 1,700 yd, in 
the shadow zone. This fluctuation of received sound 
field intensity has become the subject of special in¬ 
vestigations, which are summarized in Chapter 7. 
The principal purpose of transmission runs, however, 
is to obtain the average transmission properties of the 
ocean with a given set of oceanographic conditions. 

To obtain a representative average, it is necessary 
to select a sample of signals, assign to each signal an 
individual sound field amplitude, and then to strike 
an average. The final result of these steps, the average 
sound field intensity or sound field level, will depend 
not only on the record obtained, but also on the 
details of the sampling and averaging procedure 
employed. UCDWR has standardized these pro¬ 
cedures to insure intercomparability of results ob¬ 
tained at different times and by different research 
groups. The procedure is described in a report by 
UCDWR 5 and will be briefly recapitulated in the 
following paragraphs. 

In the selection of a sample several requirements 
must be satisfied. The sample of individual signals 
must be large enough so that the standard deviation 
of the average is not much larger than of the order 
of 1 db. Moreover, the benefit of averaging will be 
obtained only if the sample covers a period of time in 
which the transmission passes through a number of 
maxima and minima, for otherwise the average would 
be an average of individual signals most of which 
may be relatively high or relatively low. On the other 
hand, the period of time covered by the sample must 
be short enough so that it corresponds to a negligible 
change of range between the two ships and a negligible 
change in the large-scale temperature structure. 

The standard procedure for supersonic work, de¬ 
signed to strike a compromise between these require¬ 
ments, has been to select five signals, equally spaced 
during a period of 20 sec. Since the standard devia¬ 
tion of an individual signal from average intensity is 
between 2 and 4 db in most samples, the standard 
deviation of the arithmetical average of five signals 
from the average of a very large number of signals is 
between 1 and 2 db, (1/V n — 2 times the standard 
deviation of the individual signals). 

At WHOI, the rule has been to use as a sample ten 
consecutive signals. Since signals are transmitted 
about 1.2 sec apart, a sample extends over a period of 
12 sec. This method, although slightly different from 
that employed at UCDWR, leads to averaged ampli- 



Figure 13. Signal with high noise background. 



















TRANSMISSION RUNS 


83 





Figure 14. Signal with end spikes. 


tudes which differ from those obtained by the other 
method, but probably by no more than the internal 
spread of either method. 

The next step consists of the assignment of an 
amplitude to each member of the selected sample. If 
the received signal were square-topped, like the 
emitted signal, this step would raise no questions. 
However, received signals are often far from square- 
topped. It was decided at UCDWR to assign to each 
signal the peak value of the amplitude registered any¬ 
where during one signal, with two qualifications. One 
concerns noise received simultaneously with the 
signal. At low signal levels, the noise which is re¬ 
ceived continuously shows up as a very striking 
“spiny” record (shown in Figure 13). Such spines 
superimposed on the signal are disregarded. This 
rule presupposes that noise spikes can be distin¬ 
guished from rapid signal fluctuations. It has been 
found that all persons competent to evaluate record 
film are able, with but little practice, to make that 
distinction. The other qualification concerns “end 
spikes.” Frequently, there is interference between 
sound traveling via two different routes, for example, 
direct and surface-reflected sound. As the two paths 
do not have exactly the same length, the intensity at 
the beginning and end of the signal may be markedly 
different from the intensity during the signal. In the 
case of destructive interference, the signal then as¬ 
sumes the shape shown in Figure 14. The end spikes 
appearing in such signals are also disregarded. 

The rules just outlined have certain advantages 
and certain drawbacks. The principal advantage is 
that the peak amplitude of a signal can be read much 
more rapidly than such quantities as mid-signal 
amplitude; also, it is unambiguous. 1 he drawbacks 
appear when the signal envelope is not smooth. In 


that case the sound emitted during a short signal 
interval arrives at the receiver during a much longer 
period of time, with the result that the energy re¬ 
ceived during an interval equal to the signal length 
is substantially less than all the energy received. This 
effect will lie quite conspicuous for very short signals, 
but negligible for continuous transmission (10-sec 
signals). As a result, the average amplitude is very 
definitely a function of ping length, when peak 
amplitudes are used; it very likely would not be a 
function of ping length if the amplitudes of individual 
signals were defined in a different manner. One 
possible solution has been suggested by the group 
which is carrying out transmission experiments at 
WHOI. They have constructed an integrating cir¬ 
cuit. If the received signal is squared and fed directly 
into this integrating circuit, the recording instru¬ 
ment shows the total energy received. This would be 
strictly proportional to the signal length and would 
thus provide a measure typical for the ocean and its 
overall transmission properties. Any deviation from 
strict proportionality would be indicative of non¬ 
linear transmission and would, therefore, be of the 
greatest importance. At the time of this writing, no 
such experiments had been carried out. 

Once individual amplitudes have been assigned to 
the five signals that comprise one sample, the average 
amplitude is found by taking the arithmetical mean 
of the five individual amplitudes. This procedure has 
the advantage of simplicity. Alternatively, one could 
compute the mean level or the mean intensity 
(squared amplitude). A very rough estimate shows 
that in a typical record the averaging of amplitudes 
and of intensities will lead to results which are dif¬ 
ferent by about 1 db. While this difference depends 
on the assumed distribution function of amplitudes, 













84 


EXPERIMENTAL PROCEDURES 


it is not likely to be a dominant cause of error in a 
determination of the transmission loss. 

Transmission work at single sonic frequencies 
began only recently, and the analysis procedure has 
not yet been very well standardized. Records ob¬ 
tained up to the present appear to indicate that 
fluctuation of signal intensity is much less severe at 
sonic frequencies than at supersonic frequencies. On 
the other hand, because of image interference, sys¬ 
tematic changes in signal level are observed at short 
ranges which vary so rapidly with range that any 
averaging procedure would obscure them. For this 
reason, in transmission work at frequencies from 200 
to 1,800 c individual signal levels rather than sample 
averages are reported. 

In the records obtained at sonic frequencies the en¬ 
velope of the signal trace, as a rule, is badly serrated. 
The fuzziness of the envelope is probably caused by 
the unfavorable signal-to-noise ratio, which is about 
1 db for 1.8 kc and lower, somewhat higher for 
22.5 kc, and by the relative narrowness of the filters 
used in the recording channels. If random noise is 
received through a wide filter, the oscillograph trace 
has a typical “spiked” appearance, that is, the noise 
is characterized by sudden high peaks of short dura¬ 
tion. If the filter is narrow, as it must be in sonic 
transmission work, the individual peaks are lowered 
and broadened, and their separation from the 
single-frequency signal is more difficult. For this 
reason, the person reading the film record does not 
attempt to measure the “peak” level, which would 
be fictitious, but estimates and reports the average 
amplitude. It has been found that the uncertainty 
introduced by this estimate is less than 1 db, on 
the average. 

The final step in the processing of a transmission 
run consists of the recording of the computed signal 
intensity. Since in most transmission runs the range 
is altered by a factor of 10 to 100, the signal levels 
change in the course of a run by a large number of 
decibels. It has, therefore, been found useful not to 
plot signal level in decibels below transducer output 
directly, but to take out the bulk of the variability 
by plotting the transmission anomaly. Usually, the 
transmission anomaly is plotted as the ordinate down¬ 
ward, with range as the abscissa. Theoretically, this 
curve should pass through zero for zero range. In 
view of the great experimental difficulties involved in 
the determination of the signal level at very short 
ranges, the curves usually stop at a range of 100 yd 
or more. 


4.4 ECHO RUNS 

As mentioned before, transmission runs are by far 
the most important useful method of obtaining in¬ 
formation on the propagation of sound in the ocean. 
The other two methods, which are of secondary im¬ 
portance, will be discussed in the next two sections 
for the sake of completeness. 

Echo runs have been carried out both on specially 
designed standard bodies and on chance targets, such 
as wrecks, in order to study the dependence of echo 
level on range and in order to study fluctuation and 
coherence. The principal purpose of echo runs has 
usually been to study not the propagation of sound 
between echo-ranging transducer and target, but 
rather the effect of certain targets on the received 
echo. (See Chapters 18 to 26 of this volume.) 

The equipment used in echo runs differs from that 
used in transmission runs in that sending and re¬ 
ceiving stacks are aboard the same ship and the same 
sound head is used both for sending and receiving. A 
change-over relay connects the sound head first with 
the sending stack and then, immediately following 
the emission of the signal, with the receiving stack. 

Artificial targets have been developed for research 
and training purposes. Natural targets usually re¬ 
flect very differently at different aspects; most arti¬ 
ficial targets are designed to minimize this kind of 
directionality without sacrificing too much overall 
reflecting power. There is one geometrical shape 
which remains the same regardless of any twisting of 
the cable from which the target is suspended. That is 
the sphere. From the point of view of constant re¬ 
flecting power, spheres constitute ideal artificial 
targets. 

Unfortunately, the reflecting power of a sphere, 
while constant, is fairly small. To obtain useful 
echoes from spheres at distances similar to the ranges 
commonly encountered in practical echo ranging, one 
would have to use spheres with a diameter greater 
than 30 ft. It was found, however, that a 10-ft sphere 
was almost unmanageable at sea. The only spheres 
which could be handled with ease were spheres with a 
diameter of 2 or 3 ft. 

In the search for an artificial target with a large 
target strength, the best solution found so far has 
been the triplane 6 ’ 7 shown in Figure 15, which com¬ 
bines ease of handling with a reflecting power com¬ 
parable to that of a submarine. It is a well-known 
fact, sometimes used in optical signaling, that a ray 
which has been reflected from three planes which are 



INFORMATION OBTAINABLE FROM REPORTED RANGES 


85 


mutually perpendicular leaves in a direction exactly 
opposite to the incident direction. The action of a tri¬ 
plane can, therefore, be compared with that of a 
single plane perpendicular to the incident rays. That 
is why a triplane reflects a larger percentage of the 
incident energy back into the transducer than any 
other body of equal size. 



Figure 15. Triplane. 

Another type of artificial target is the so-called 
echo repeater. This is a device which acts essentially 
as a relay. It consists of a transducer with power out¬ 
put proportional to the incident sound energy re¬ 
ceived by a hydrophone. Echo repeaters have been 
used only for training purposes. A full description of 
the echo repeater can be found in two UCDWR re¬ 
ports. 89 

4.5 INFORMATION OBTAINABLE FROM 
REPORTED RANGES 

The research methods described in Sections 4.3 and 
4.4 of this chapter are of comparatively recent origin. 
During the first year of the war, the only information 
available consisted of observed maximum echo and 
listening ranges obtained by surface ships on escort or 
patrol duty and by research vessels on ocean cruises. 

In testing the performance of echo-ranging gear, 
several workers recognized the strong variability of 
achieved maximum echo ranges. 101112 Vessels at¬ 


tached to the West Coast Sound School at San Diego 
found in practice maneuvers that ranges in the after¬ 
noon compared unfavorably with ranges in the morn¬ 
ing of the same day. These observed maximum ranges 
gave the first clue to the existence of shadow zones in 
the presence of sharp downward refraction. Maxi¬ 
mum ranges obtained by echo ranging on submarines 
above and below the depth of the layer revealed the 
existence of a layer effect (see Section 5.3.3). Also, the 
tabulation of observed ranges over various types of 
ocean bottom in shallow water gave clues as to the 
effect of the bottom on sound transmission. 

Because of the unexplained variability of observed 
maximum ranges, it was decided to set up the investi¬ 
gation of the underwater sound field as a research 
program, and the quasi-laboratory methods of trans¬ 
mission runs and echo runs were developed. Since the 
inception of the transmission program, observed echo 
ranges have rarely served as scientific evidence; they 
have continued to serve as a stimulus for the investi¬ 
gation of new problems and as signposts on the road 
to solutions. The SS Nourmahal, a converted yacht 
with a deep projector and with unusually quiet 
machinery, has reported extreme echo ranges in the 
presence of very deep isothermal layers; as a result, 
the sound field in deep mixed layers was investigated. 
To give another example, earlier experience had 
shown that the sound field in shallow water over 
MUD bottoms is very nearly the same as the sound 
field in deep water, because MUD reflects sound 
rather poorly. Unexpectedly long echo ranges in 
certain areas in which the bottom was classified as 
MUD led to a new program aimed at a differentia¬ 
tion of the various bottom sediments now called 
MUD. Also, data obtained at WHOI seem to show 
that attenuation increases at very high wind forces. 

These observed ranges have been obtained both by 
naval vessels in regular operations and by research 
vessels. A number of naval vessels have sent to WHOI 
records of observed maximum echo ranges along with 
bathythermograph slides. Additional observed ranges 
were obtained by research vessels on extended cruises 
in various parts of the world. The range data thus 
obtained do not permit any detailed conclusions con¬ 
cerning the transmission loss as a function of range 
but serve to indicate whether sound transmission was 
good, fair, or poor. A summary of the theory of 
maximum echo ranges is presented in Volume 7 of 
Division 6. 






Chapter 5 


DEEP-WATER TRANSMISSION 


T ransmission in deep water, where bottom- 
reflected sound is unimportant, is somewhat 
simpler to study than transmission in shallow water. 
Even when the effects of the bottom have been elimi¬ 
nated, however, sound transmission in the ocean re¬ 
mains an exceedingly complex phenomenon. The 
theoretical results based on the elementary ray 
theory and on an idealized ocean stratified in uni¬ 
form horizontal layers are seldom realized exactly in 
the sea. Sometimes this simple picture leads to er¬ 
roneous results, even qualitatively. Moreover, the 
only constant element in underwater sound trans¬ 
mission is change. No one ping resembles the preced¬ 
ing. In this chapter, the rapid fluctuation of trans¬ 
mitted sound from one second to the next is ignored 
and reference is made throughout to averages based 
on many consecutive pings. However, even these 
averages sometimes vary considerably. 

Although the theory developed in the previous sec- 
tions is admittedly imperfect and may be incorrect in 
principle, this theory is nevertheless retained as the 
framework on which to hang the discussions of the 
observational material. The theory is believed valu¬ 
able, partly in indicating which results may be ex¬ 
pected to have general validity beyond the particular 
conditions under which the results were obtained. 
Even more important, a discussion of the interrela¬ 
tion between facts and theories is essential for an in¬ 
telligent formulation of research programs. In the 
long run, progress in any scientific problem can be 
achieved most efficiently by formulating hypotheses 
and then testing them in critical experiments. To lay 
the groundwork for such future research is, in large 
part, the objective of the present chapter. 

5.1 FACTORS AFFECTING 

DEEP-WATER TRANSMISSION 

In principle, the propagation of sound can be com¬ 
pletely determined if the nature of the medium 
through which the sound passes is known. In the 
present section a description is given of the known 
properties of the sea which are believed to influence 
underwater sound transmission. 


5.1.1 Meaning of “Deep W ater’” 

For the purposes of this chapter, water is deep 
when the bottom has a negligible effect on under¬ 
water sound propagation. From a theoretical stand¬ 
point this has the very simple meaning that the bot¬ 
tom is ignored; the ocean is thought of as extending 
to infinite depths. From the observational stand¬ 
point, this means that only those observations will 
be considered here on which the bottom is believed 
to have no effect. 

In general, the bottom can have several effects on 
underwater sound. Sound energy reaching the bottom 
may be partly reflected back at various angles into 
the body of the sea and partly transmitted into or 
absorbed by the bottom. The relative amounts re¬ 
flected and absorbed depend on the depth and the 
nature of the bottom, prevailing refraction condi¬ 
tions, and sea state. This dependence and, generally, 
the effect of the bottom on sound transmission will 
be discussed in detail in Chapter G. Furthermore, the 
presence of the bottom affects the background. Some 
of the sound reflected backward by the bottom 
reaches the receiver and gives rise to a ringing sound 
known as reverberation. 

For most of the observations discussed in this 
chapter, short pulses of sound are used. With this 
technique, sound that has traveled to the bottom and 
has been reflected toward the hydrophone can readily 
be distinguished from sound that has traveled directly 
from projector to hydrophone. Thus, for most obser¬ 
vations the bottom-reflected sound can readily be 
distinguished. If the bottom is rough, an appreciable 
amount of sound may reach the hydrophone after 
having been scattered from various portions of the 
sea bottom so that the signal is followed by a long 
single-frequency train of forward reverberation. 
Usually, the direct signal is so far above this back¬ 
ground of scattered sound that forward reverbera¬ 
tion is negligible in the evaluation of transmission 
observations. 

It is of importance to know when the ocean is ef¬ 
fectively deep in practical applications of underwater 
sound. With present echo-ranging gear, an echo from 


86 


FACTORS AFFECTING DEEP-WATER TRANSMISSION 


87 


o 

Nl 

X 

»— 

CL 

Ui 

O 


Ui 

2 


X 

o 

< 

UJ 


< 

5 


Id 

o 

< 


o 

cc 

£ 



1000 


2000 


3000 4000 5000 

DEPTH IN METERS 


6000 


7000 


0000 


Figure 1. Distribution of depths in the sea. 


a submarine or typical surface vessel cannot ordi¬ 
narily be detected more than 3,000 yd from the pro¬ 
jector in water of any depth, except under unusually 
favorable conditions. All supersonic projectors are 
highly directional with not much energy radiated at 
angles more than G degrees from the axis. If the bot¬ 
tom is more than 150 fathoms below the projector, 
and the water is isothermal, very little of the energy 
in an echo-ranging pulse will reach the bottom at 
ranges less than 3,000 yd or return to the surface at 
ranges less than 6,000 yd. Thus, the echo from targets 
near maximum range will contain very little bottom- 
reflected sound; however, the background for such 
echoes may contain some bottom reverberation. II 
sharp temperature gradients are present in the upper 
layer of the ocean, the sound beam will be bent down 
more sharply, and a considerable amount ot bottom- 
reflected sound could reach a target 3,000 yd away in 
water 150 fathoms deep. The bottom reverberation 
in such conditions may be quite intense even at 1,500 
yd. To insure that bottom-reflected sound cannot re¬ 
turn an echo in practical echo ranging, a depth of 
more than 200 fathoms is required, while twice this 
depth is required to eliminate bottom reverberation. 
For various types of tilting beam equipment, sound 
scattered from the bottom can be important even in 


somewhat deeper water. For most echo-ranging situa¬ 
tions, however, 100 or 150 fathoms is a more repre¬ 
sentative dividing line between deep and shallow 
water. 

It makes very little difference whether the point of 
division is taken as 150 fathoms, or 100 fathoms, as 
has been done in the manuals of echo-ranging predic¬ 
tion issued by the Navy, 1 " or 200 fathoms, as has 
been suggested. Water depths between 100 and 1,500 
fathoms are quite uncommon. Figure 1 shows the 
distribution of depths in the sea. 3 It is evident from 
Figure 1 that almost all of the ocean bottom is either 
less than 100 fathoms, about 200 meters, below the 
surface, or more than 1,500 fathoms, about 3,000 
meters, below the surface. 

Water which is deep for echo ranging may be 
shallow for sonic listening, since average listening 
ranges are so much longer than average echo ranges, 
and since sonic listening gear is nondirectional. Lis¬ 
tening ranges are often greater than 10,000 yd. Ex¬ 
cept in the deepest parts of the ocean, sound arriving 
from such long ranges will contain bottom-reflected 
sound. Sonic gear is usually nondirectional in a verti¬ 
cal plane, at least at low frequencies, and bottom-re¬ 
flected sound in 2,000 fathoms may contribute ap¬ 
preciably to the received signal. Thus, for sonic lis- 
































88 


DEEP-W ATER TRANSMISSION 



Figure 2. Typical temperature-dopth curve. 


tening at long ranges the ocean is rarely if ever deep. 

Supersonic listening gear, on the other hand, is 
usually sharply directional in the vertical plane and 
will discriminate against bottom-reflected sound from 
ships in the same way that it will discriminate against 
bottom-reflected echoes. Thus water which is deep 


for supersonic echo ranging will also be deep for 
supersonic listening, even though supersonic listening 
ranges may exceed sonic ranges. 

Evidently, at supersonic frequencies most of the 
ocean is effectively deep for practical purposes. Even 
at low sonic frequencies, transmission in water deeper 


















































FACTORS AFFECTING DEEP-WATER TRANSMISSION 


89 


than 1,500 fathoms is practically deep-water trans¬ 
mission under many conditions. Thus the study of 
sound transmission in deep water is of considerable 
practical importance. 

5 . 1.2 Vertical Temperature Structure 
aiul Computed Ray Diagrams 

The temperature distribution in the ocean largely 
determines the sound velocity distribution, which we 
have seen is an important factor in sound intensity. 
For this reason, measurement of ocean temperatures 
at various depths has been an integral part of the re¬ 
search on sound transmission and is also important 
in the tactical use of sonar equipment. 

The temperature in the ocean is affected by the 
absorption of radiation from the sun and sky, by the 
cooling of the surface layer by evaporation, by dis¬ 
placements due to currents and upwelling, and by the 
addition of fresh water near shore. Usually a water 
column in the deep sea can be divided into three 
principal layers, shown by the sample temperature- 
depth plots in Figure 2: (1) a relatively warm surface 
layer, which is subject to daily and seasonal changes 
in thickness and vertical temperature gradients, (2) a 
layer of transition at mid-depths called the thermo¬ 
cline, in which the temperature decreases rapidly 
with depth, and (3) the cold deep-water layer, in 
which the temperature decreases only gradually with 
depth. A detailed discussion of the temperature dis¬ 
tribution in the ocean is given in Volume 6 of Divi¬ 
sion 6. Here, only a few of the basic temperature- 
depth patterns are described and their expected in¬ 
fluence on underwater sound transmission briefly dis¬ 
cussed. 

It will be pointed out in later sections that the 
transmission loss is least and sound ranges are longest 
when the surface layer is reasonably isothermal and 
deeper than about 100 ft. Such deep isothermal layers 
tend to occur when the water at the surface is losing 
more heat than it is gaining, as in midwinter in the 
high latitudes. The colder surface water will be 
heavier than the water just beneath and will mix 
with it. As a result, a surface layer of more or less 
constant temperature will be formed. In midwinter 
the isothermal surface layer is usually several hun¬ 
dred feet deep, except in tropical waters, where this 
depth varies from 50 to 500 ft depending on ocean 
currents and other factors. In very high latitudes the 
isothermal layer may extend down to the ocean bot¬ 
tom in February or March. 


The ray diagram for sound transmission in the case 
of an isothermal surface layer above a thermocline 
has approximately the characteristic shape shown 
in Figure 3. According to the simple ray theory, the 
sound beam should split at the bottom of the isother¬ 
mal layer, with the upper portion bending gradually 

TEMPERATURE 

PROJECTOR 

DEPTH 

1 

Figure 3. Ray diagram for isothermal water above 
thermocline. 




back to the surface because of the effect of pressure 
and the lower portion bending sharply down into the 
thermocline. Temperature-depth patterns resulting 
in such a predicted ray diagram are called “split- 
beam” patterns. If the intensity of the sound field 
were measured along the vertical line SS' in Figure 3, 
the intensity immediately below the isothermal layer 
should decrease sharply with increasing depth. After 
having reached a minimum, the sound field intensity 
should begin to increase slowly with increasing depth 
until finally the edge of the main sound beam is 
reached. 

At longer ranges with split-beam patterns a meas¬ 
urement of intensity, as along the vertical line RR' in 
the diagram, should indicate a substantial amount of 
sound in the isothermal layer, but in or below the 
thermocline very little sound should appear. As 
shown in Section 5.3.2, these predictions of theory 
for long range are not confirmed by the observations, 
which show no clear trace of the predicted shadow 
boundary below the layer. 

It may be pointed out that the shaded area in 
Figure 3 is usually not a true shadow zone, even in 
theory. The temperature-depth graph usually curves 
continuously from the isothermal layer into the 
thermocline, rather than breaking at the sharp angle 
shown in Figure 3. As a result of this curvature, some 
direct sound always theoretically penetrates the 
“shadow zone” in this case, but this theoretical sound 
is much weaker than the sound actually observed. 

Heating of the surface layer of the ocean some¬ 
times produces a temperature gradient which extends 
all the way to the sea surface. Such conditions are 
most common at high latitudes during the summer 
months, when the surface water is gaining more heat 







90 


DEEP-WATER TRANSMISSION 


than it is losing. Surface heating and marked temper¬ 
ature gradients in the top 30 ft of the ocean are par¬ 
ticularly marked on summer afternoons with calm 
seas and cloudless skies. At night, or during periods of 
high winds, gradients near the surface tend to dis¬ 
appear. 

The ray diagram computed for a temperature 
gradient extending up to the surface is shown in 
Figure 4. The decrease of sound velocity with in- 

TEMPERATURE 

PROJECTOR 

DEPTH 

1 


Figure 4. Ray diagram for negative gradient extend¬ 
ing to surface. 

creasing depth bends the entire sound beam down¬ 
ward, as discussed in Chapter 3. Beyond a certain 
limiting range, which increases with increasing depth, 
no sound ray can penetrate, and a shadow zone of 
complete silence should result. Observations of under¬ 
water sound transmission confirm the presence of this 
shadow zone when the temperature gradient is suf¬ 
ficiently strong, about i degree or more in the top 
30 ft. The only sound reaching such a shadow zone is 
the scattered sound, which also produces reverbera¬ 
tion back at the echo-ranging projector. When the 
surface gradients are weak, however, other effects 
become important, and shadow zones do not appear. 

In addition to these two basic but simplified tem¬ 
perature-depth patterns, innumerable varieties of 
intermediate situations occur. Complicated tempera¬ 
ture structure is especially likely in the surface layer; 
and the accurate computation of a ray diagram from 
a temperature-depth record can be very laborious. 
Since the observations usually do not confirm the 
detailed predictions of the simple theory, which is 
based on small details of the temperature structure, 
the computing of ray diagrams for these intermediate 
cases is of limited usefulness. 

Sharp positive temperature gradients are extremely 
rare in deep water. Such gradients are stable only 
when accompanied by positive salinity gradients. 
Salinity gradients may also affect sound velocity, 
but their effect is usually negligible compared to that 
of temperature gradients. Salinity gradients may be 
appreciable in some near-shore areas, where large 
rivers drain into the sea, and at the coastwise margins 



RANGE-■- 



of the permanent ocean currents, such as the Gulf 
Stream. In such regions, sharp positive temperature 
gradients may occur. In the open ocean, however, 
they are usually less than a few tenths of a degree in 
30 ft. Because of 1 he rarity of sharp positive gradients, 
there is a complete absence of data on sound trans¬ 
mission in deep water through regions of strong up¬ 
ward refraction. 

5.1.3 Variability of Vertical 
Temperature Gradients 

The way in which ocean temperature changes with 
depth is variable from time to time and from place to 
place. Gradual changes from day to day and from one 
geographical region to another have an important 
effect on the performance of sonar gear. These changes 
are discussed in detail in Volume G of Division 6, and 
form the basis for the Sound-Ranging Charts 4 and the 
Submarine Supplements. 5 

In some areas these changes are so rapid that they 
greatly complicate the study of underwater sound 
transmission. In the coastal waters off San Diego, a 
bathythermograph lowered at one end of a transmis¬ 
sion run frequently showed marked differences from 
the bathythermogram obtained at the other end, with 
wholly different ray diagrams resulting. Two samples 
of such records are presented in Figure 5. Some of this 
variation represents a change with time, while much 
of it arises from changes with location. In early com¬ 
parisons between transmission data and the com¬ 
puted range to the shadow boundary, an average was 
taken of the ranges computed from several bathy¬ 
thermograph records. More recently, a single bathy¬ 
thermograph record taken on the receiving vessel has 
been used at UCDWR in studying the relation be¬ 
tween the transmitted sound intensity and the tem¬ 
pera t ure-depth record. 

Temperature Microstructure and Effects 

In addition to these large temperature changes 
over several thousand yards, smaller changes take 
place over much smaller distances. These changes 
may affect the way in which the sound beam travels 
through the water. In Chapter 3 the predictions of the 
ray theory were discussed for a sound beam passing 
through an ocean in which the sound velocity depends 
only on depth, but decreases gradually with depth. 
In such an ideal ocean an exact temperature-depth 
record would be similar to that shown in Figure 6. A 
plot of temperature against range at any depth would 








FACTORS AFFECTING DEEP-WATER TRANSMISSION 


91 


give the horizontal lines shown in the figure. Under 
such temperature conditions a shadow zone is pre¬ 
dicted at a certain limiting range. 

In practice, the ocean is never stratified in plane 
parallel layers, each of uniform temperature. In¬ 
stead, an exact temperature-depth record might be 



50 60 70 


TEMPERATURE IN DEGREES F 

Figure 5. Temperature conditions at beginning and 

end of transmission run. 

similar to that shown in Figure 7. Plots of tempera¬ 
ture against range at different depths would be curves 
similar to the wavy curves also shown. This “tem¬ 
perature microstructure” must be taken into account 
in any explanation of observed underwater sound 
transmission. 

The evidence available on thermal microstructure 
is very limited. Measurements on a surface ship are 
difficult to interpret because of the rise and fall of the 
measuring instrument through the water. Some of 
this vertical motion arises from the roll and pitch of 
the measuring ship, and some from the distortion of 
the temperature-depth pattern by the surface waves. 
For this reason, a very small-scale microstructure is 
difficult to measure from a surface ship, although 
changes over a hundred yards or so can usually be 
disentangled from the more rapid changes resulting 
from roll and pitch. Measurements from a submarine 
show conclusively the presence of complicated ther¬ 
mal microstructure. 6 In Figure 11 oi reference 0, 
fluctuations of the vertical gradient are shown which 
amount to about 0.020 degree per ft over patches 
about 100 yd long. This result was obtained with 


large temperature gradients present near the surface. 
When the bathythermograph shows mixed water to 
more than 100 ft, the microstructure is much less 
marked. 

o 


£ 50 

UJ 

u. 
z 

X 
F 

& 100 
Q 

150 

Figure 6. Temperature distribution in ideal ocean. 



/ 



7 






70 F 


65 F 


60 F 


RANGE 


60 70 

TEMPERATURE F 


VERTICAL SECTION OF OCEAN 


A general theory of underwater sound transmission 
which takes microstructure into account has not yet 
been formulated. However, certain general results 
seem apparent. These temperature fluctuations are 
usually fairly small compared to the smoothed gradi¬ 
ent, and on the whole the actual temperature-depth 



TEMPERATURE F 

Figure 7. Temperature distribution in actual ocean. 


SOURCE 



- MICROSTRUCTURE ABSENT 

- MICROSTRUCTURE PRESENT 

Figure 8. Distortion of sound beam by microstruc¬ 
ture. 

pattern portrayed in Figure 7 does correspond to the 
ideal pattern shown in Figure 6. Thus some corre¬ 
spondence may be expected between observed sound 
transmission data and predictions based on the 
smoothed temperature-depth pattern. 


































































92 


DEEP-WATER TRANSMISSION 


The chief effect of temperature microstructure is 
to introduce irregularities into the path of the indi¬ 
vidual sound rays. They will be slightly bent away 
from the average ray path in random fashion, as in¬ 
dicated in Figure 8. Considering a sound beam as a 
whole, we may expect that microstructure will very 
slightly broaden the beam pattern, although such 
broadening effects have never been determined with 
assurance. Within the sound field, local intensities 
may show deviations from the average values which 
would be observed in the absence of microstructure; 
these local deviations will be discussed in Chapter 7. 
Another effect of these irregularities is that sound 
may penetrate with a small but observable intensity 
into regions which are shadow zones according to the 
large-scale ray pattern. 

It is possible to estimate the effect which micro- 
structure will have on the ray trajectories of indi¬ 
vidual sound rays. Theoretical analysis shows that 
with certain simplifying assumptions the rms lateral 
displacement Ay of a sound ray because of micro¬ 
structure is given by the formula 7 

Ay = WbGtf. (1) 

In this equation G is the rms value of the fractional 
velocity gradient caused by microstructure, R is the 
range, and b is a quantity having the dimension of a 
length, which may be called the patch size of the 
microstructure. Roughly speaking, b is the average 
distance over which the vertical velocity gradient 
caused by microstructure retains the same sign. To 
derive this formula, an expression was first obtained 
for the lateral displacement of a ray passing through 
a given microstructure. This expression was then 
squared and averaged. The square root of the final 
result gave expression (1). 

It has already been noted that fluctuations of the 
vertical temperature gradient, amounting to 0.02 F 
per ft over patches about 100 yd in length, have been 
reported. If these values for G and for b are substi¬ 
tuted into equation (1), it is found that at a range of 
1,000 yd the rms lateral spreading of the sound beam 
amounts to about 20 ft; while at 2,500 yd it amounts 
to 70 ft and at 4,000 yd at 150 ft. These figures indi¬ 
cate that at these ranges random spreading of the 
transmitted sound beam, because of microstructure, 
will obscure bending of the sound rays due to large- 
scale vertical temperature structure if the vertical 
gradient is of the order of 0.1 F in 30 ft. Actual obser¬ 
vation shows that even negative gradients of four 
times this magnitude often fail to produce clearly 


recognizable shadow zones, although the sound does 
weaken gradually with increasing range. It is not 
known at present whether microstructure will fre¬ 
quently have a magnitude appreciably in excess of 
that assumed for the estimate of lateral beam spread. 
If not, some other cause must be invoked for an ex¬ 
planation of why weak negative gradients do not 
produce shadow zones. 

Since no complete theory exists at the present time 
capable of explaining in detail the results obtained in 
transmission runs, much of the discussion of under¬ 
water sound transmission must be empirical in char¬ 
acter. It is possible, for example, that some of the 
empirical relationships found between the smoothed 
temperature-depth curves and the measured trans¬ 
mission anomalies result primarily from an oceano¬ 
graphic correlation between the temperature micro- 
structure and the smoothed distribution of tempera¬ 
ture with depth. Such observed empirical relation¬ 
ships are valuable, but until their basic physical cause 
is explained they should be used with caution since 
they may be valid only for the particular time and 
place in which the observations were made. 

5.1.4 Classification of Batlivther- 
niogranis 

For practical use of temperature-depth informa¬ 
tion some simple method of classifying bathythermo¬ 
graph records is essential. Even if the predictions of 
ray theory were exactly fulfilled, practical require¬ 
ments would probably rule out the time and effort 
required to construct ray diagrams and to compute 
theoretical intensities. Thus, a set of rules has been 
devised to classify temperature-depth records by the 
properties which are acoustically significant. 

Such classifications have also proved useful in 
transmission research. Since the simple ray theory 
was clearly inadequate, some other basis was required 
for comparing measured anomaly curves with the 
corresponding bathythermograms. In view of the 
complexity of possible temperature-depth curves, no 
classification can be entirely satisfactory. All such 
classifications must be regarded as preliminary until 
sufficient acoustic information is available to indicate 
exactly what features of the temperature-depth pat¬ 
tern are significant in any situation. 

Present systems of classification are primarily de¬ 
signed to correspond to different types of transmis¬ 
sion loss for a shallow projector, about 15 ft. When 



FACTORS AFFECTING DEEP-WATER TRANSMISSION 


93 


the temperature increases with depth sufficiently for 
the temperature at some depth below the projector 
to be greater than the projector temperature, rays 
leaving the projector at slight downward inclina¬ 
tions will in theory be bent back up to the surface 
again; the transmission anomaly for a shallow hydro¬ 
phone should therefore be low, although experimental 
data on this point are lacking. When such positive 
gradients are present, the temperature pattern is 
called positive, sometimes denoted by PETER. 

Such patterns may be more completely character¬ 
ized by the depth of the layer of maximum tempera¬ 
ture and the difference between maximum tempera¬ 
ture and the temperature at projector depth. The 
sharpness of the underlying thermocline may also be 
acoustically significant. 

Other temperature-depth records are classified by 
the temperature difference in the top 30 ft. If this 
difference is 0.3 F or less, the water is said to be 
isothermal, and the temperature pattern is called 
mixed, sometimes denoted by the word MIKE. When 
this difference is greater than 1/100 of the surface 
temperature the computed range to the shadow 
boundary is less than 1,000 yd, for projector at 15 ft, 
hydrophone at 30 ft. For this temperature condition, 
the predicted shadow zone is commonly observed, 
and transmission to a shallow hydrophone becomes 
poor for ranges greater than 1,000 yd. Such a tem¬ 
perature distribution is called a sharp negative pat¬ 
tern, sometimes denoted by NAN. Temperature dif¬ 
ferences intermediate between MIKE and NAN 
tend to be somewhat variable and are classified as 
weak negative or changing patterns, denoted by 
CHARLIE. 

One exception is included in this relatively simple 
scheme. When the temperature difference from 0 to 
30 ft is large enough to give a NAN pattern, but the 
temperature difference from 15 to 50 ft is 0.2 F or 
less, the pattern is classified as CHARLIE. With such 
an extremely shallow and negative gradient and with 
the projector in isothermal or nearly isothermal 
water, good but variable sound conditions may be 
expected. This type of pattern is the most favorable 
for the formation of a sound channel. 

With MIKE and CHARLIE patterns the depth 
and sharpness of the thermocline would be expected 
to affect the transmission of sound to a deep hydro¬ 
phone. Appropriate methods for characterizing these 
quantities are discussed in Section 5.3, where the 
acoustic measurements made with a hydrophone in 
or below the thermocline are summarized. 


Another more 

detailed system of classification, 

which supplements the classification of negative 
gradients into MIKE, CHARLIE, and NAN pat¬ 
terns, has been devised at UCDWR. This system 
utilizes the depths at which the temperature is 0 . 1 , 
0.3, 1.0, 5.0, and 10 F below the surface temperature. 

These depths are 
and D- 0 . 

called, respectively, Z>i, Z) 2 , D 3 , Z) 4 , 

For statistical analysis, these depths are given code 
numbers between 0 and 9 by the following numerical 

scale. 


Code digit 

Depth D in feet 

0 

1 

0 SD< 5 

5 < D < 10 

2 

10 ^ D < 20 

3 

20 ^ D < 40 

4 

5 

40 ^ D < 80 

80 < D < 160 

6 

160 ^ D < 320 

7 

320 ^ D 

9 

D greater than greatest 
depth reached by bathy¬ 


thermograph 


Any bathythermogram may then be coded by giv¬ 
ing the code digits corresponding to Di, D 2 , D 3 , Z) 4 , 
D 5 . The surface temperature T is also coded by giving 
T /10 to the nearest whole number and by placing this 
digit after the other five and separating it by a deci¬ 
mal point. The code numbers for D x through D- 0 and 
also T /10 are denoted by di, d 2 , d 3 , d it d 3 , and d 6 , re¬ 
spectively. The series of numbers is then written as 
d 1 d 2 d 3 d 4 d 5 .de, as for example 23 457.6. The accurate 
determination of dj is very difficult because of the 
wide trace made by the bathythermograph near the 
surface; since the variability of this small tempera¬ 
ture difference will usually be high, there is some 
question whether this quantity is usually significant . 

Examples of bathythermograms classified by the 
two methods are given in Figure 9. These two systems 
of classification supplement each other and should 
probably be used together. The code system is 
probably most useful for surface gradients, where 
considerable detail is provided. For example, it is 
shown in subsequent sections that the transmission 
of sound to a shallow hydrophone depends markedly 
on d 2 for different NAN patterns. On the other hand, 
for a fixed d 2 , transmission to a shallow hydrophone 
differs markedly between NAN and MIKE patterns. 
For deep gradients the code system is somewhat less 
useful, owing to the very expanded depth scale. For 
example it is frequently not clear from the present 
code whether a deep hydrophone is above or below 
the thermocline. It seems likely that when more com- 



DEPTH IN FEET DEPTH IN FEET DEPTH IN FEET 


94 


DEEP-WATER TRANSMISSION 


DEGREES FAHRENHEIT 



MIKE 

CODE SYM80L 66 679.7 


DEGREES FAHRENHEIT 



CHARLIE 

CODE SYMBOL 22 569.6 


DEGREES FAHRENHEIT 



DEGREES FAHRENHEIT 



MIKE 

CODE SYMBOL 44 555.7 


NAN 

CODE SYMBOL 00 156.7 


DEGREES FAHRENHEIT 



CHARLIE 

CODE SYMBOL 02 799.8 


DEGREES FAHRENHEIT 



Figure 9. Classification of bathythermograph records 






























































































































































































































































TRANSMISSION IN ISOTHERMAL WATER 


95 


plete acoustic information is available, a modification 
of the code system, more closely related to the typical 
structure of the thermocline, will prove desirable. 

5.2 TRANSMISSION IN 

ISOTHERMAL WATER 

When temperature and salinity gradients are ab¬ 
sent, the transmission of sound may be expected to be 
relatively simple. If pressure did not affect sound 
velocity, sound would travel outward in straight 
lines, and the inverse square law of intensity decay 
would be directly applicable. Since the effect of pres¬ 
sure on sound velocity is in fact very small, one may 
expect straight-line propagation to provide a reason¬ 
ably good first approximation to the actual situation. 
An examination of the ray diagram given in Figure 23 
of Chapter 3 shows that the ray leaving horizontally 
from a projector 15 ft deep in isothermal water is not 
bent up to the surface until it reaches a range of about 
1,000 yd. On the other hand, n an isothermal layer 
100 ft thick the range to the shadow boundary is al¬ 
ways greater than 2,200 yd. Thus, one may expect 
that, certainly for ranges less than 1,000 yd and prob¬ 
ably also for ranges up to 2,000 yd, the assumption 
that sound travels in straight lines in isothermal 
water is legitimate. This expectation is justified by 
the observations. It will be shown later that the as¬ 
sumption of straight-line propagation agrees with 
some of the data out to very much greater ranges 
than might be expected. The reason for this is not 
known. In the following discussion, sound transmis¬ 
sion measurements in an isothermal surface layer of 
the ocean will, therefore, be discussed as though the 
sound rays did, in fact, travel in straight lines in 
such a layer. 

Even with all sound rays traveling in straight 
lines, two influences act to disturb the ideal inverse 
square law discussed in Chapter 2. In the first place, 
sound is reflected from the sea surface; in the second 
place, various impurities, and possibly also the water 
itself, absorb energy passing through the sea, convert¬ 
ing this energy to heat. The effects of surface reflec¬ 
tion and absorption on the transmission of sound are 
discussed later. 

5 . 2.1 Image Effect 

It was shown in Section 2.6.3 that sound reflected 
from the surface can reduce the sound intensity close 
to the surface to a very small value. This effect arises 
from the phase reversal suffered by a wave when it is 


reflected at a free surface. If p\ is the pressure ampli¬ 
tude at 1 yd from the source, and if hi and /i 2 are the 
depths of source and receiver respectively, we see 
from equation (129) of Chapter 2 that the pressure 
amplitude at the range R is given by 

a r+ i 2 Pi • 9 h\h% 

Amplitude = — sin 2ir - 

R R\ 

If the simple inverse square law for intensity were 
satisfied, the pressure amplitude at the range R 
would be proportional to 1 /R, that is, 


Amplitude = —' 


The transmission anomaly, which is the transmission 
loss in decibels above that predicted by the inverse 
square law for intensity, is therefore given by 


A = 


— 20 log 


2 sin 2-7T 


hjh 

R\. 


( 2 ) 


If the surface is assumed to reflect only a fraction 
yl of the sound energy incident on it, the analysis in 
Section 2.6.3 must be modified. With a little mathe¬ 
matical manipulation, the transmission anomaly for 
this case becomes 


A = —10 log 


hihi , 
1 — 2y a cos 4 tt — + yl 


(3) 


The transmission anomalies resulting from this for¬ 
mula for different values of y a are plotted in Figure 
10 . 

This analysis is of doubtful validity for short wave¬ 
lengths since neglect of the surface water waves in 
heavy seas is probably not legitimate for sound waves 
only a few inches long. With calm seas, however, the 
interference patterns predicted by the above analysis 
have occasionally been observed at 24 kc at close 
ranges. Equation (3) must be used with caution for 
ranges much greater than 1,000 yd. The upward re¬ 
fraction caused by the pressure effect, as well as the 
variation in travel time caused by thermal micro- 
structure, distort and obscure the interference pat¬ 
tern predicted by the elementary theory. However, 
the exact limits of validity of equations (2) and (3) 
must be determined empirically. 

The available data show that for sufficiently low 
frequencies, equation (2), or equation (3) with an 
amplitude reflection coefficient y a nearly equal to 
unity, provides an approximate description of the ob¬ 
served transmission. In particular, beyond a range 
R', corresponding to a path difference of a half wave¬ 
length, the transmission anomaly increases steadily 







96 


DEEP-WATER TRANSMISSION 



0.1 0.2 0.3 0.5 0.7 I 2 3 5 7 10 


_ R 

4h t h 2 

Figure 10. Theoretical transmission anomalies for different values of the reflection coefficient of the surface. 


from its minimum value and, beyond a range of about 
2 R', increases equally with 20 log R. This range R' is 
given by the equation 


More specifically, equation (2) is applicable for 
frequencies of less than 200 c. For this low frequency, 
this equation may be used out to ranges of 1,000 to 
2,000 yd, beyond which bottom-reflected sound, even 
in 2,000 fathoms, is usually stronger than the direct 
sound. At 600 c the correspondence between theory 
and observation is not so good, although the general 
tendency predicted by equation (3) is definitely 
present; possibly the best value of y a for 600-c sound 
is around 3d> to %. At frequencies greater than several 
thousand cycles, no definite trace of image effect has 
been consistently observed in the open sea. 

One of the earliest sources of observational infor¬ 
mation on this subject consisted of a set of transmis¬ 
sion runs made jointly in 1943 by the Columbia Uni¬ 
versity Division of War Research at the U. S. Navy 
Underwater Sound Laboratory, New London 
[CUDWR-NLL], University of California Division 
of War Research [UCDWR], and Massachusetts 
Institute of Technology Underwater Sound Labora¬ 


tory [MIT-USL], 8-10 An acoustic minesweeper was 
used as the source, and the signal was received with 
band-pass filters centered at different frequencies. 
The water depth was 600 fathoms. Unfortunately, 
the temperature near the surface was not isothermal 
to 100 ft; in fact, in some of the runs sharp negative 
gradients extended to the surface, and for some runs 
the hydrophone was below a sharp thermocline. Since 
the image effect was shown consistently at 250 and 
700 c at ranges between 100 and, 300 yd, where bend¬ 
ing by temperature gradients rarely affects measured 
sound intensities, the data may be taken as an indica¬ 
tion that the same effects would also appear in 
isothermal water. 

Some transmission runs at 2 kc and at 8 kc also 
showed some leveling off of the transmission anomaly 
at higher ranges. 8 However, the steeper slope was 
never so marked at ranges less than 500 yd that it 
could be attributed to image effect rather than to 
downward refraction. 

A detailed comparison between theory and obser¬ 
vation for the sound of lower frequencies is made in 
reference 10. The theory takes bottom-reflected sound 
into consideration and achieves rather good agree¬ 
ment with the observational data. However, the 
bottom-reflected sound comes in at such close range 



































TRANSMISSION IN ISOTHERMAL WATER 


97 


that the results do not cast much light on the prob¬ 
lems of transmission of low-frequency sound in deep 
water at ranges greater than a few hundred yards. 

A much more complete set of measurements on low- 
frequency transmission in deep water is given in a 
progress report from UCDWR. 11 That report sum¬ 
marizes the first results obtained in a long-range pro¬ 
gram designed to investigate sound transmission at 
frequencies of 200, GOO, 1,800, 7,500, and 22,500 c. 
Since short pulses of sound were used in this work, 
the direct and bottom-reflected sound can be distin¬ 
guished by the difference in travel time, provided the 
range is not too great. Also, specially designed trans¬ 
ducers were employed with the result that the power 
output was high and usually remained constant dur¬ 
ing each transmission run. 

Sample results for individual transmission runs at 
200, 600, and 1,800 c are shown in Figure 11. As is 
evident from the bathythermograph code given with 
each plot, these data were obtained with isothermal 
water at the surface. Each point in these plots repre¬ 
sents the transmission anomaly found for a single 
sound pulse. The curved lines represent the values 
found from equation (3) with the amplitude reflec¬ 
tion coefficient y a chosen to give the best fit to the 
observations. For each frequency, the measured 
transmission anomalies at different ranges are moder¬ 
ately accurate relative to each other, while the abso¬ 
lute values are less reliable; hence each set of ob¬ 
served anomalies in Figure 11 lias been shifted verti¬ 
cally to give the best agreement with the theoretical 
curves. 

The agreement between the plotted points and the 
theoretical curves is typical of the results generally 
obtained in deep water with an isothermal layer. At 
200 c, image effect is usually marked, and at 1,000 to 
2,000 yd it reduces the sound level about 15 db on the 
average below the inverse square value; this corre¬ 
sponds to an effective reflection coefficient y a of 0.8, 
somewhat lower than the value of 0.9 for the 200-c 
curve in Figure 11. At 600 c, the effect is less marked, 
corresponding to an effective reflection coefficient y a 
in the neighborhood of 0.7, and an average transmis¬ 
sion anomaly of only about 10 db at 1,000 yd. How¬ 
ever, the dip in intensity shown in Figure 11 at about 
60 yd, in agreement with theoretical prediction, sug¬ 
gests that image effect is in fact present. At 1,800 c, 
the predicted minimum between 150 and 200 yd is ap¬ 
parently present, but the reduction in intensity at 
ranges between 1,000 and 2,000 yd is quite small, 
corresponding to a value of about 0.5 for y a . Thus at 


frequencies above about 1,000 c, it appears that image 
effect is relatively unimportant. This is in general 
agreement with theoretical expectations. 12 At ranges 
greater than 2,000 yd, bottom-reflected sound is 
usually dominant, even in water several thousand 
fathoms deep. 

The reflection coefficients to which the different 
curves in Figure 11 correspond should not be taken 
as a measure of the amount of sound reflected by the 
surface. It is, of course, virtually certain that most 
of the sound reaching the ocean surface is reflected 
back into the ocean in some direction, except possibly 
when strong winds produce absorbing bubbles close 
to the surface. However, some of this sound may be 
reflected in directions quite different from the sound 
reflected at an ideally flat, horizontal surface. Also, 
the relative phases of the direct and surface-reflected 
sound may be altered by the irregularities in the 
ocean surface. As a result of these two factors, the 
image effect to be expected for a flat, perfectly re¬ 
flecting surface may be modified. Equation (3) is 
then useful as a semi-theoretical, semi-empirical 
formula for fitting the observed data. 

A somewhat different manner of presentation, 
which includes all the data available at the time refer¬ 
ence 11 was written, is shown in Figure 12. Here, all 
available anomalies at certain fixed ranges are plotted 
on a linear range scale. In such a plot, the short ranges 
are too compressed to show the interference patterns 
characteristic of the image effect. However, such plots 
are very suitable if emphasis on the data at longer 
ranges is desired. 

These data provide supporting evidence for the 
general statements made in the discussion of Figure 
11 . The rise of the median curves for 600 and 1,800 c 
is probably not real, but simply a result of observa¬ 
tional selection; at the long ranges, the received sig¬ 
nals are difficult to distinguish from noise, and only 
those few signals can be measured which, because of 
fluctuation, rise far above the noise level. Thus the 
median curves in Figure 12 are probably considerably 
higher at long range than they would be if all the runs 
yielding data at short range could have been con¬ 
tinued successfully to long range. 

5 . 2.2 Absorption 

At frequencies above several thousand cycles, im¬ 
age effect is usually unimportant, and in the absence 
of temperature and salinity gradients, absorption be¬ 
comes the chief effect modifying the inverse square 




TRANSMISSION ANOMALY IN DB 


98 


DEEP-WATER TRANSMISSION 





SLANT RANGE IN YAROS 

Figure 11. Typical transmission anomalies at sonic frequencies, source at 14 feet, hydrophone at 50 feet. 






















































































-10 

0 

10 

20 

30 

10 

0 

10 

20 

*10 

0 

10 

20 , 


TRANSMISSION IN ISOTHERMAL WATER 


99 



• ••• 

• • 

• • • 

• •• 

••• 



• • — 

• • *1 




A • 

«»»\ 

• \ • • • • 

• \ • 

\ • 

\* # * 

• a* — 

• • 

i • • 

• • • 

• 

• •• 

• • 

-• 

* * 

• • 

• • 

• 

/ 

lAl. 





• •* 

• 


• • 

i • • • 

— ••• • 

• 

• -TfZ 



**** -».- 





• • \ 


• 1 

h— •••• 

• • 




• \ • 

• • • • • • • 






• 







• 

* • ••• 

• 


600 CYCLES 


• • 

• 

• 

• • • 

• 


(36 RUNS) 


. 

• •• • 






• 

• 








Figure 12. Average transmission anomalies at sonic frequencies. 






















































100 


DEEP-WATER TRANSMISSION 


law of simple geometrical spreading. A detailed 
analysis was given in Section 2.5 for the absorption 
resulting from viscosity. However, it was pointed out 
in that section that the observed absorption of sound 
in the ocean is far greater than can be accounted for 
by viscosity. 

The effect of absorption on underwater sound trans¬ 
mission is shown most simply by considering the 
propagation of sound in an unbounded homogeneous 
ocean. Let .7 be the total energy proceeding outward 
from a sound source in each second. If the ocean were 
not at all absorbing, this same amount of energy 
would spread to all ranges. If the source is nondirec- 
tional, this energy would be spread over an area of 
iirR 2 at a range R, and the sound intensity 7 would be 
given by the equation 


I = 


./ 

iirR ' 1 


F 

R 2 


(5) 


where F, defined in Chapter 2, is given by 


F = 



In the presence of absorption, a constant fraction 
n of the sound energy is absorbed in each yard of 
sound travel. Thus in a distance dr, an amount of 
energy imiF dr will be absorbed per second, and con¬ 
verted into heat energy. The constant n is called the 
absorption coefficient of the water. The decrease Air dF 
of the sound energy over this distance will equal the 
energy absorbed, or 4 irnF dr. Thus we have the 
equation 


dF 

dR 


-nF, 


( 0 ) 


which has the familiar exponential solution 


F = F 0 e~ nR 


(7) 


The sound intensity is then given by the equation 


I = 



( 8 ) 


In terms of decibels, this equation may be written 

10 log I = 10 log F 0 — 20 log R -— (9) 

where a/1000 = lOn logic e = 4.34a; a expresses the 
absorption in decibels per kiloyard and is called the 
coefficient of absorption. The transmission loss H, 
as defined in Chapter 4, is 10 log F 0 — 10 log 7. The 
transmission anomaly A is simply 77 — 20 log R. 


Thus we have the simple equation 


A = 


aR 
1000 ’ 


( 10 ) 


Hence, in an unbounded medium, absorption pro¬ 
duces a transmission anomaly which increases lin¬ 
early with the distance covered by the sound beam. 

Scattering of sound is more complicated than ab¬ 
sorption. When scattering rather than absorption is 
present, equation (10) may still be used to describe 
the decay of the unscattered sound. However, the 
sound that has been scattered must also be considered 
in computing the expected sound intensity. It is 
shown in Section 5.4.1 that scattering is probably not 
very important in isothermal water. However, since 
the exact role of scattering in isothermal water is not 
certain, and since some forms of scattering may be 
very important when temperature gradients are 
present near the surface, it is customary to refer to 
the combined effects of absorption, scattering, and 
similar phenomena as attenuation. The quantity a, 
determined by direct measurement of A and use of 
equation (10), is then called the coefficient of attenua¬ 
tion. Attenuation, as so defined, includes all effects 
which may produce a transmission anomaly. 

Extensive observations at a number of laboratories 
indicate that in isothermal water the transmission 
anomaly does, in fact, increase linearly with increas¬ 
ing range, in accordance with equation (10).Thus, the 
attenuation coefficient a for each frequency is a con¬ 
stant for any one run. The data at 24 kc, in a report 
on attenuation issued by UCDWR, provide a check 
of this point. 13 Of the many runs available in deep 
water off the coast of southern California and Lower 
California, at the time reference 13 was written, 65 
were made when the temperature difference from the 
surface to a depth of 30 ft was 0.1 F or 0.0 F. For all 
these runs the graphs of transmission anomaly against 
range could “reasonably be approximated by straight 
lines beyond a range of 1,000 yards.” Two sample 
plots of transmission anomaly, with the corresponding 
temperature-depth records, are shown in Figure 13. 
Each point represents the average amplitude of five 
different pings. 

The linearity of the observed points is evident in 
Figure 13. On the average, about half of the plotted 
points lay within 2 db of the straight-line curve 
drawn for each run. Thus it is reasonable to conclude 
that in water which is isothermal from the surface to 
30 ft, the transmission anomaly increases linearly 
with range from 1,000 to more than 6,000 yd. Since 




TRANSMISSION IN ISOTHERMAL WATER 


101 


o 


600 


-20 


a> 

o 


0 


RAY DIAGRAM 




SOUND FIELD DATA 


4840 4890 4940’ 

SOUND VELOCITY IN FT PER SEC 



0 2000 4000 6000 


DATE 

2- 28-1944 

TIME 

1705 

BT CLASS MIKE 

WATER 

DEPTH 2 200 FM 

SEA 

2 

SWELL 

3 

WIND 

FORCE 4 



RANGE IN YARDS 

Figure 13A. Sample transmission anomaly in isothermal water. 


only about half the runs were made with shallow 
hydrophones, between 16 and 30 ft, and the other 
half with deeper hydrophones, usually below the 
thermocline, this result apparently applies for sound 
transmitted below the thermocline as well as for 
sound in the isothermal layer. This linearity of the 
transmission anomaly for deep hydrophones is dis¬ 
cussed again in Section 5.3.2. 

It is perhaps surprising that the transmission 
anomalies should be straight lines out to long ranges 
when the isothermal layer is at most a few hundred 
feet thick. In a completely isothermal layer the un¬ 
ward bending of the sound, caused by the increase of 
pressure with depth, should give rise to a shadow 
zone near the surface at 3,000 yd for a thermocline 
starting at 150 ft below the projector and at 6,000 yd 
for one starting at a depth of 600 ft below. While 
sound reflected from the surface would penetrate this 
shadow zone computed for the direct sound, some 
drop in transmission might nevertheless be expected 
at the shadow boundary. 

It is possible that slight negative gradients of 
about 0.1 F in 30 ft were present in most of these 


measurements since slight gradients are common off 
San Diego and are very difficult to measure exactly 
with the bathythermograph. Such a slight gradient 
would offset the effect of pressure on sound velocity 
and give nearly straight-line propagation out to 
considerable range. 

The observed results could also be qualitatively 
explained on the assumption that the temperature 
in the isothermal layer is not completely constant, 
but varies irregularly from point to point. It was 
shown in Section 5.1.3 that the microstructure ob¬ 
served in regions of sharp temperature gradient can 
broaden the sound beam in the vertical direction by a 
hundred feet in several thousand yards. It micro- 
structure of similar effectiveness were present in the 
isothermal layer, this alternate up-and-down bending 
from microstructure would wash out the pressure 
effect entirely and would enable some direct sound 
to travel to an indefinite range in the isothermal layer. 
Since a fraction of this sound would be bent down into 
the thermocline at all ranges, the linearity of the 
transmission anomaly curve at depths below the 
thermocline might also be explained on this basis. 











































102 


DEEP-WATER TRANSMISSION 


RAY DIAGRAM 


BT INFORMATION 




SOUND VELOCITY IN FT PER SEC 


DATE 

1 1 - 27 - 194 3 

TIME 

1830 

BT CLASS MIKE 

WATER 

DEPTH 2000 FM 

SEA 

2 

SWELL 

3 

WIND 

FORCE 3 



Figure 13B. Sample transmission anomaly in isothermal wat/T. 


However, since there are no extensive measurements 
on small-scale thermal structure in the isothermal 
layer, no conclusions about these problems can be 
reached at the present time. 

The straight line of best fit drawn on a plot of 
measured transmission anomaly against range gives, 
with moderate precision, the attenuation coefficient 
for the time and place of the measurements. Because 
the transmission anomaly usually approximates a 
straight line for measurements at fixed depth in 
isothermal water, the probable error of the attenua¬ 
tion coefficient found in this way is only about db 
per kyd, for runs extending out to about 6,000 yd. 

The detailed variation of the attenuation coefficient 
in isothermal water from place to place and from 
time to time has not been thoroughly explored; and 
so it is most useful to deal with an average attenua¬ 
tion coefficient. The evidence on variation of trans¬ 
mission loss in isothermal water will be given in Sec¬ 
tion 5.2.3. 

The most complete investigation of the average 
attenuation coefficient at 24 kc in isothermal water is 
apparently that presented in a report by UCDWR. 14 


CD 



Figure 14. Average transmission anomalies in iso¬ 
thermal water. 


All deep-water runs in isothermal water are analyzed 
together. This analysis shows that if the water is es¬ 
sentially isothermal to about 100 ft the measured 
transmission anomalies do not depend on the tem¬ 
perature at greater depths or on the state of the sea 
surface, at least out to ranges of 3,000 yd. 

Transmission runs made under these conditions 
have been combined to yield the average curves of 


























































TRANSMISSION IN ISOTHERMAL WATER 


103 



0 1000 2000 3000 


RANGE IN YAROS 

Ftoure 15. Individual transmission anomalies in isothermal water. 


transmission anomaly shown in Figure 14. Only runs 
with a hydrophone at 16 ft have been included. In the 
curve for 24 kc, taken from reference 14, runs made 
with a total temperature change of less than 0.3 de¬ 
gree between the surface and 80 ft were included. 
Since a general analysis shows that temperature dif¬ 
ferences as small as 0.2 degree in the top 30 ft do not 
affect transmission anomalies appreciably, probably 
much the same curve would be obtained if only those 
runs in water isothermal to less than 0.1 F were con¬ 
sidered. In the curve for 60 kc (taken from Volume 7 
of Division 6), all runs made with an approximately 
isothermal surface layer were included. The slopes of 
the two average curves in Figure 14 give attenuation 
coefficients of 4.0 db per kyd at 24 kc and 13.5 db 
per kyd at 60 kc. 

In Figure 15 are plotted all the individual points 
used at 24 kc, with solid lines connecting the median 
points and dashed lines connecting the upper and 
lower quartiles. The average quartile deviation shown 
in Figure 15 is 2.6 db. This is about the same as the 
deviation of any single transmission anomaly curve 
in isothermal water from a straight line, and is also 
about the same as the experimental error, discussed 
in Chapter 4, in the determination of the transmis¬ 
sion loss from the average of five observations. 

Other sets of measurements give similar values of 
the attenuation coefficient a. Many transmission runs 
were made some time before the war by XRL. u 
In these runs, the range was commonly opened from 
less than 1,000 to more than 10,000 yd. These data 


were originally plotted in terms of intensities rather 
than transmission anomalies. It was found that in a 
considerable number of runs, plots of sound intensity 
(in decibels) against range gave essentially straight 
lines beyond 1,000 yd. The measurements were re¬ 
analyzed and the results reinterpreted in terms of 
transmission anomalies. 13 

For the data reported in reference 16, all the 
straight-line graphs were obtained when the water 
was isothermal (to +0.1 C) from the surface to more 
than 100 ft. The average attenuation coefficients 
found at 17.6, 23.6, and 30 kc were 1.8, 4.4, and 6.5 
db per kyd, respectively. Half the observed values 
lay within ± 1 db per kyd of these average values. 

The NRL and UC-DWR measurements described 
above are the only ones in deep water which have 
been analyzed in terms of the temperature structure 
present at the time the measurements were made. A 
considerable body of other measurements have been 
made to determine the attenuation coefficient, but 
these are less reliable. 

Measurements of bottom-reflected sound in shallow 
water have been used to determine the attenuation 
coefficient resulting from absorption and scattering 
in the volume of the ocean. In particular, sound has 
been sent out vertically, and the strength of the echo 
received in different depths of water used as a meas¬ 
ure of attenuation in the water. Since these measure¬ 
ments depend entirely on the reflection coefficient of 
the bottom, they can give results on attenuation 
only if it is assumed that the reflection coefficient of 

























104 


DEEP-WATER TRANSMISSION 



12 5 10 20 50 10 0 2 0 0 500 1000 

RANGE IN YARDS 


Figure 16. Measured and computed intensities in shallow water. 


the bottom is independent of depth and location. 
There is no evidence for this assumption, especially 
since the nature of the bottom in the deep ocean is 
believed to be somewhat different from that in shal¬ 
low water. Moreover, the attenuation coefficient 
measured at considerable depths has no necessary 
connection with the coefficient in the surface layers of 
the sea. Thus, these data are of little use, except for 
predicting the levels of sound reflected vertically 
from the bottom in different depths of water. 13 

Values of the attenuation coefficient have also been 
found from measurements of horizontal transmission 
in shallow water. Since these depend on the numerical 
value of the bottom-reflection coefficient, they do not 
give an accurate indication of the attenuation in deep 
water of constant temperature. However, at high 
frequencies, the error introduced into the results by 
an incorrect value of the bottom-reflection coefficient 
becomes small, and these values are relatively trust¬ 
worthy. 

An investigation along these lines is described in a 
report issued by UCDWR. 18 In this work, sound was 
transmitted from a dock in San Diego harbor through 
water 30 to 50 ft deep. As the range wis changed, 
from 10 to 1,000 yd, the recorded sound intensities 
fluctuated rapidly, since the direct, surface-reflected, 
and bottom-reflected rays interfered, first construc¬ 
tively, then destructively. The exact computation of 
these interference effects would be very complicated. 
Instead, only the highest peaks reached by the 


fluctuating sound were considered; for these peaks it 
was assumed that the different rays involved were all 
in phase, all interfering constructively. These meas¬ 
ured values were then compared with theoretical 
curves, computed without regard for absorption. 
These curves were found by adding the calculated 
direct sound to the calculated sound from all the 
images resulting from successive surface and bottom 
reflection; the directivity of the sound projector was 
taken into account, and all the different rays were 
assumed to arrive in phase. The difference between 
the observed and computed curves was then used as 
a measure of the absorption. A sample plot showing 
the observed peaks of the sound intensity and the 
theoretical curves for different values of y a , the 
amplitude reflection coefficient of the bottom, is 
given in Figure 16 for a sound frequency of 100 kc. 
For comparison, the inverse square curve expected in 
a deep, ideal ocean is also shown in the figure. Un¬ 
fortunately, no temperature measurements were made 
during this work. Measurements made in the same 
location one year later showed that the temperature 
gradients were usually small because of strong tidal 
currents. The temperature difference between 0 and 
20 it was found to be less than 0.2 F, 94 per cent of 
the time, and less than 0.1 F, 75 per cent of the time. 
Thus, it is probably legitimate to take the attenua¬ 
tion coefficients of this study as representative of 
isothermal or nearly isothermal water. 

At 60, 80, and 100 kc, this work gave attenuation 

























































TRANSMISSION IN ISOTHERMAL MATER 


105 



FREQUENCY IN CYCLES 


Figure 17. Dependence of attenuation coefficient on frequency. The points on the curve are from the following refer¬ 
ences: □ NRL, 13 and 16; V UCDWR, 14; O UCDWR, 18; • CNRC, 19; • Fresh Water, 20; ■ Fresh Water, 22; 
A Fresh Water, 23; A WHOI, Chapter 9 of this book. 


coefficients of 18, 26, and 32 db per kyd, respectively, 
for an assumed amplitude reflection coefficient of the 
bottom of 0.5 (energy loss of 6 db per reflection), cor¬ 
responding to the SAND-AND-MUD bottoms over 
which the measurements were made. A change of the 
reflection coefficient by 0.2 in either direction changes 
the attenuation coefficient by about 2.5 db per kyd 
in the same direction; this variation may be taken 
as a rough estimate of the probable error of the re¬ 
sults. Owing to this high probable error, the values of 
2.0 and 7.0 db per kyd, found at 24 and 40 kc re¬ 
spectively, are of relatively low weight and may be 
disregarded. 

At frequencies between 500 and 2,500 kc, extensive 
measurements of attenuation have been made by the 
Canadian National Research Council. 19 A projector 
was mounted on a dock in Vancouver Harbor in 13 to 
25 ft of water. The receiver was also mounted on the 
same dock at distances varying up to 100 ft. As a 
result of the high directivity of the projector, surface 
and bottom-reflected sound were largely eliminated. 
The slope of the transmission anomaly was measured 


to give an attenuation coefficient at each frequency. 
Relative probable errors of these coefficients, esti¬ 
mated from the reproducibility of the results, 
averaged about 7 per cent. No temperature measure¬ 
ments were made. Over such short ranges any gradi¬ 
ents would have had a negligible effect. 

No measurements at frequencies above 3 me are 
available for sound in the ocean. However, such 
measurements have been made in the laboratory.- 0- - 3 
Those of reference 20 extend down to 2.8 me, where 
the values found are of the same order of magnitude 
as those determined in the ocean. Other determina¬ 
tions of absorption in fresh water in the frequency 
range between 200 and 4,000 kc are about ten times 
as high as those found in the sea. 24-26 These fresh¬ 
water measurements are not in good agreement with 
each other and may be affected by systematic errors. 
Since the sea-water values taken from reference 16 
were made over a much greater sound path, these 
should be much more reliable, and in any case, consti¬ 
tute better evidence for the attenuation of sound in 
the sea; the fresh-water measurements in references 

































































































































































106 


DEEP-WATER TRANSMISSION 




TIME OF DAY 

Figure 18. Variability of attenuation coefficient during one day. 


24, 25, and 26 may therefore be ignored in the present 
discussion. 

At frequencies below 14 kc no data on deep-water 
attenuation in the surface layers of the sea are avail¬ 
able. The long-range measurements of explosive 
sound propagation, discussed in Chapter 9, give an 
upper limit on the attenuation coefficient deep in the 
ocean. Although the data are uncertain, the results 
quoted in Chapter 9 indicate that the attenuation at 
2,000 c is probably less than 5 X 10 -2 db per kyd. 

These different determinations of the attenuation 
coefficient are combined in the plot of a against fre¬ 
quency shown in Figure 17. The dashed line gives a 
curve of best fit drawn through the plotted points. 
The solid line in this figure gives the value of a to be 
expected from viscous damping, taken from Section 
2.5. Evidently, at frequencies above 1 me the at¬ 
tenuation coefficient is three to four times the classi¬ 
cal value. In a UCDWR memorandum, 27 this dis¬ 
crepancy is attributed to an additional viscous force 


proportional to the rate of compression of the water. 
Such a force has usually been neglected in hydro¬ 
dynamics, since ordinarily water flows like an in¬ 
compressible fluid. Although no tests of such an 
hypothesis have been suggested, it is entirely possible 
that this “compressional viscosity” may be respon¬ 
sible for the observed values of a at frequencies above 
1 me. 

No explanation has yet been advanced for the ob¬ 
served values of the attenuation at somewhat lower 
frequencies. Since all the measured values between 
10 and 100 kc were obtained in temperate latitudes in 
water well above the freezing temperature, it is possi¬ 
ble that these values are not applicable for all oceano¬ 
graphic conditions. It is still not wholly certain that 
the attenuation observed for supersonic frequencies 
in isothermal water is entirely the result of absorp¬ 
tion rather than scattering; however, the weakness of 
scattered sound observed for backward scattering 
(reverberation) and for forward scattering (incoher- 















































TRANSMISSION IN ISOTHERMAL M ATER 


107 


ent sound measured in shadow zones) makes this 
highly probable. 

At frequencies below 15 kc the attenuation of 
sound is largely conjectural. The shallow water meas¬ 
urements discussed in Chapter 6 show that a is not 
greater than about 1 db per kyd at frequencies below 
2 kc. On the other hand, if the attenuation in this 
frequency range were as low as 0.1 db per kyd, high¬ 
speed warships could be heard consistently many 
hundreds of miles away. Since such listening ranges 
are apparently not obtainable, it may be inferred 
that the attenuation coefficient in the surface layers 
of the ocean is greater than 0.1 db per kyd at all sonic 
frequencies. Such a high value is not necessarily in¬ 
consistent with the much lower value observed in the 
deep sound channel since the attenuation at low 
frequencies near the ocean surface may result pri¬ 
marily from scattering of sound out of the isothermal 
layer into the thermocline, where it is bent sharply 
downward, and is lost. 

5.2.3 Variation of Transmission Loss 

The previous section has discussed average values 
of the attenuation coefficient at each frequency, but 
has ignored changes in this coefficient, ft has already 
been noted that from a single run out to 0,000 yd the 
attenuation coefficient can be determined with a 
probable discrepancy of only about Yi db per kyd 
from its true value at that particular time and place. 
Since the scatter of the observed values exceeds this, 
it may be inferred that the attenuation coefficient in 
sea water is probably not constant. 

In the measurements reported in reference 16, for 
example, half the attenuation coefficients at 24 kc 
differed by more than 1 db from the average value of 
4.4 db per kyd. Corresponding variations also ap¬ 
peared at the other two frequencies (17.6 and 30 kc). 
However, those runs in which more than one fre¬ 
quency was used show a good correlation (correlation 
coefficient r between 80 and 85 per cent) in the varia¬ 
tion of the coefficients for the three frequencies. 

A good illustration of this correlation is provided 
by the runs made during one 24-hr period (February 
14-15, 1944), analyzed in reference 13. Seven meas¬ 
urements of the vertical temperature structure were 
made during this period. In each case, no variation of 
temperature of more than 0.2 I was noted down to 
depths of 120 ft, but this was also the limit of ac¬ 
curacy of the thermometer on these days. The surface 
temperature changed appreciably during the period, 


however, probably as a result of the changing position 
of the vessels. The variation of the attenuation 
coefficients for sound at 17.6, 23.6, and 30 kc is 
plotted in Figure 18 with the measured surface tem¬ 
perature. Evidently the attenuation coefficients at 
the three frequencies changed very substantially; 
however, the difference in the attenuation coefficients 
between the different frequencies was more nearly 
constant. 

Under some conditions, however, the attenuation 
coefficient during a 48-hr period is less variable. A 
series of transmission measurements was made by 
UCDWR in the deep water off Point Conception, 
California, where a persistent well-mixed layer was 
to be expected. Measurements were carried out dur¬ 
ing and after a storm, with winds of force 3 to 6 
(Beaufort scale). The surface layers of the sea were 
probably better mixed during these transmission runs 
than for any other reported transmission experiments. 
A typical temperature-depth record taken during 
these measurements is shown in Figure 19. 



54 56 58 60 62 64 66 


WATER TEMPERATURE IN DEGREES F 

Figcrf. 19. Depth record for Point Conception runs. 

The cumulative distribution of attenuation coeffi¬ 
cients for the data taken with the shallow and deep 
hydrophone is shown in Figure 20. These are plotted 
on probability paper, so that a normal, or Gaussian, 
distribution of plotted points would lie on a straight 













10S 


DEEP-WATER TRANSMISSION 


line. For the data shown in this figure, half of the ob¬ 
served values of a fall within about 0.(3 db per kyd of 
the average values. This is so close to the probable 
error of 0.5 db per kyd for a single determination of a 
that the result may be entirely due to observational 
errors. Certainly the reduced scatter can be attrib¬ 
uted to the relatively uniform conditions prevailing 
during these tests. The observed attenuation coef¬ 
ficients for the deep hydrophone will be discussed in 
Section 5.3. In addition, the relatively small scatter 
shown in Figure 15, about the same as the observa- 



I 5 20 40 60 80 95 99 99.9 

PER CENT OF ALL ATTENUATIONS 
GREATER THAN THE VALUE PLOTTED 


Figure 20. Distribution of attenuation coefficients 
for Point Conception runs. 


suggests that in isothermal water the attenuation 
may be relatively constant. Further information is 
required, however, before any very definitive conclu¬ 
sions can be drawn about the variability of the at¬ 
tenuation coefficient when the temperature of the 
water in the upper 100 ft of the sea is approximately 
constant. Most of the UCDWR data have not been 
analyzed with this purpose in mind. In reference 13, 
where a high variability of a is found, the runs in 
isothermal water are not treated separately. Also, 
some of the runs included do not extend out very far; 
when marked peaks in the anomaly curve are present, 
the attenuation coefficient for such runs may be as 
low as 1 db per kyd. An examination of a sample run 
of this type, shown in Figure 21, indicates that such 
values are not necessarily indicative of the attenua¬ 
tion over longer ranges. Thus, these data do not cast 
much light on the variability of the attenuation 
coefficient in isothermal water. 


It is tempting to assume that the values shown in 
Figure 17 represent pure absorption, and that any 
variations from these values found in approximately 
isothermal water represent distortion of wave front 
by temperature gradients too small to be detected 
reliably on the bathythermograph. More accurate 
data on transmission in mixed water and more ac¬ 
curate thermal measurements would be required to 
test such an hypothesis. 

5.2.4 Short-Range Transmission 

The goal of transmission studies is to relate the 
sound intensity at any range to the sound output of 
the source. Most transmission measurements at sea, 
however, do not measure the sound level closer than 
about 100 yd from the source. In principle it should 
be possible to measure the absolute sound level in the 
water, and also to measure the absolute level 1 yd 
from the projector; in practice, the measuring equip¬ 
ment has apparently not been sufficiently stable to 
make these absolute measurements possible. Thus 
transmission measurements give only relative sound 
levels and may be used to give the sound level at long 
range relative to the level at several hundred yards. 
The methods used in computing transmission anom¬ 
alies are discussed in some detail in Chapter 4. For 
most of the data discussed here, the transmission 
anomaly at short range, usually about 100 or 200 yd, 
has been taken equal to zero. Thus, to find true 
transmission anomalies at long range requires infor¬ 
mation on the true value of the transmission anomaly 
at ranges around 100 yd. 

Since refraction can be ignored at such close ranges, 
the transmission anomaly for the sound passing di¬ 
rectly from projector to hydrophone must be very 
close to zero. If the surface were perfectly flat, sur¬ 
face-reflected sound would, on the average, double 
the sound intensity at ranges of several hundred 
yards, provided that the intensity is averaged over 
the interference pattern discussed in Section2.6.3;the 
transmission anomaly A at these ranges would then 
be —3 db. Sound intensity measurements between 
1 yd and several hundred yards would then show a 
gradually decreasing transmission anomaly as the 
range increased and as surface-reflected sound ap¬ 
proached the same average strength as the direct 
sound. 

However, the sea surface is never perfectly flat, 
and this fact may be expected to alter the simple 
relationships to be expected for a flat surface. Al- 

























TRANSMISSION THROUGH A THERMOCLINE 


109 




SOUND FIELD DATA 


4890 4940 4990 

SOUND VELOCITY IN FT PER SEC 









+ 

- 














DATE 11-26 

- 1943 

TIME 0800 

8T CLASS MIKE 

WATER DEPTH 

I200FM 

SEA 

2 

SWELL 

3 

WIND FORCE 

3 



4000 6000 8000 10,000 12,000 14,000 

RANGE IN YARDS 

Figure 21. Sample transmission anomaly out to short range. 


though practically all the sound which strikes the 
surface will be reflected back into the water, its 
direction will usually be affected by the water waves 
on the surface. A glance at sunlight reflected from 
the ocean surface shows how a sound beam may be 
reflected in a variety of directions at a rough surface. 
It is possible, for example, that the surface reflects 
sound predominantly downward, with little surface- 
reflected sound reaching a shallow hydrophone at 
ranges of several hundred yards or more. While some 
observational and theoretical studies of this problem 
have been attempted, the transmission anomaly at 
several hundred yards is still uncertain. One would 
expect theoretically that the anomaly might lie any¬ 
where from 0 to —3 db. This corresponds to an un¬ 
certainty of 6 db in computed echo levels from targets 
of known target strength. This important gap in 
transmission information will presumably be filled in 
when more transmission measurements have been 
made with the help of one of the several calibration 
methods discussed in Section 4.3. In most of the runs 
made up to 1944, however, neither the instrumenta¬ 
tion nor the calibration procedure was completely 


reliable. For example, the observed values of the 
anomaly at 500 yd vary from —6 db to +15 db 
when the water is isothermal to a depth of at least 
40 ft. Thus, one may readily believe that the absolute 
values of the transmission anomaly for the majority 
of the runs available may be systematically in error 
by as much as 3 db. 

5.3 TRANSMISSION FROM AN 

ISOTHERMAL LAYER THROUGH A 
THERMOCLINE 

As pointed out in Section 5.1, the ocean is rarely 
isothermal to great depths. In the more typical case, 
an isothermal layer overlies a sharp negative gradient, 
or thermocline, whose top may be at a depth any¬ 
where from less than a hundred to many hundreds 
of feet. When the isothermal layer is a hundred 
feet thick or more, sound transmission above the 
thermocline is apparently independent of the depth 
or sharpness of the thermocline, at least out to ranges 
of several thousand yards. Sound transmitted to 
points in or below the thermocline maybe appreciably 





































no 


DEEP-WATER TRANSMISSION 


weakened, however. This section discusses sound re¬ 
ceived by a hydrophone in or below the thermocline; 
emphasis is placed primarily on thermoclines a hun¬ 
dred feet thick or more. Almost all the data available 
for these conditions are at a frequency of 24 kc. 
Although some observations have been madeat higher 
frequencies, especially GO kc, practically no relevant 
information is available at sonic frequencies. 

5.3.1 Echo-Ranging Trials 

The importance of the thermocline in weakening 
sound which passes through it was first shown in 
practical echo-ranging runs. The earliest and most 
extensive data of this type were collected by the 
British. 28 In each run, a surface vessel echo-ranged on 
a submarine at continually increasing or decreasing 
range. The maximum range at which echoes could be 
obtained was noted, together with the temperatures 
and salinities at fixed depth intervals. Among various 
effects produced by refraction, the most striking was 
the reduction in maximum echo range resulting when 
a submarine submerged below the thermocline. This 
reduction in range is called layer effect. 

The quantitative importance of layer effect is evi¬ 
denced by the fact that in 40 out of G8 trials reported 
in reference 28 the maximum range decreased as the 
submarine dove from periscope depth down to about 
100 ft. Of the 28 trials in which layer effect did not 
appear, all but 5 were made in water with very weak 
temperature gradient, and 3 of these 5 exceptions 
occurred in shallow water. In 12 of the G8 trials there 
was a difference of more than 9 F between the tem¬ 
perature at projector depth and the temperature at 
the top of the deep submarine. In these 12 trials the 
maximum range on the deep submarine varied be¬ 
tween 20 and 90 per cent of the range found at peri¬ 
scope depth, the average being G5 per cent. 

Similar results were obtained in echo-ranging trials 
made by the USS Semmes (AG24, ex-DD189) on four 
fleet-type American submarines. 29 Below the isother¬ 
mal layer, which was 150 ft thick, was a sharp thermo¬ 
cline, as shown in Figure 22. When the submarine 
submerged to 250-ft keel depth or deeper, the 
maximum echo range was consistently about half the 
maximum echo range observed when the submarine 
was at periscope depth. 

5.3.2 Sample Transmission Runs 

Although the detailed interpretation of these echo¬ 
ranging results involves many complicated factors, 


such as the change of reverberation level with range, 
the general explanation is that the sound intensity 
below the layer is less than above. This result is borne 
out by detailed transmission measurements. A sample 
plot of measured transmission anomalies for deep 



TEMPERATURE IN DEGREES F 

Figure 22. Temperature-depth record for Semmes 

tests (deep layer). 

and shallow hydrophones is shown in Figure 23, 
representing a run made by UC’DWR. The computed 
ray diagram is also shown. 

The signals measured to give Figure 23 were 
coherent® at all ranges, reproducing moderately well 
the outgoing pulse. Some of the weaker signals re¬ 
ceived below the layer were characterized by “rever¬ 
beration tails,” representing incoherent sound arriv- 

a A received signal is called coherent if its envelope repro¬ 
duces faithfully the outgoing pulse. An incoherent received 
signal will in general have a ragged envelope and a length in 
excess of the length of the original outgoing pulse. Since no 
received signal portrays the envelope of the outgoing signal 
completely without distortion, coherence is a question of 
degree. 



















TRANSMISSION THROUGH A THERMOCLINE 


111 



4000 

RANGE IN YARDS 




4890 4940 4990 

SOUND VELOCITY IN FT PER SEC 


DATE 3-2-1944 
TIME 1103 


WATER DEPTH 2230 FM 


SWELL_ 4 

WIND FORCE 5 + 


Figure 23. Sample transmission anomalies above and below thermocline. 


ing after the direct signal. This scattered sound 
usually appears whenever the amplification is made 
sufficiently great. Since this incoherent sound is 
relatively more prominent when sharp downward re¬ 
fraction is present, it is discussed in Section 5.4.1. 

It is evident from Figure 23 that the plot of trans¬ 
mission anomaly against range below the layer was 
approximately a straight line. This result is quite 
general, as noted in reference 13, where it was found 
that for all runs made with isothermal water in the 
top 30 ft of the ocean the transmission anomaly plots 
were approximately straight lines. This analysis in¬ 
cluded both deep and shallow hydrophones, and the 
conclusions are therefore valid for hydrophones either 
above or below the layer. 

On the basis of the simple ray diagrams shown in 
Section 5.1.2, this result is rather surprising since the 
transmission anomaly should rise to a very high 
value at the shadow boundary, as shown in Figure 23 
by the dashed line, taken from a UCDA R internal 
report. 30 This dashed line represents the anomaly 
computed by ray tracing methods, as explained in 


Section 3.4.2. An absorption of 4 db per kyd has also 
been included. The consideration of sound reflected 
from a flat ocean surface will not change the com¬ 
puted anomaly appreciably. In particular, the shadow 
boundary will not be much affected; for the isother¬ 
mal layer shown in Figure 23, rays leaving the pro¬ 
jector at an upward angle less than 2.08 degrees will, 
after reflection, become horizontal before reaching 
the bottom of the isothermal layer, while the steeper 
rays will penetrate the thermocline at ranges of not 
more than about 3,000 yd. 

Thus, to explain the straight-line anomaly curves 
found below the layer, some mechanism must be in¬ 
volved which is not included in the simple ray theory. 
Either an irregular sea surface or thermal micro¬ 
structure explains the results qualitatively since 
either mechanism will take sound from the isothermal 
layer at all ranges and deflect some of it sufficiently 
sharply so that it will pass out of the surface layer 
into the thermocline below. At present, it is not possi¬ 
ble to state whether either mechanism can explain 
the facts quantitatively. 


















































112 


DEEP-WATER TRANSMISSION 


5.3.3 Average Layer Effeet 

Differences in transmission anomaly between shal¬ 
low and deep hydrophones are characteristic of 
isothermal water overlying a thermocline. Some evi¬ 
dence for a correlation of this difference with oceano¬ 
graphic conditions has been found. These results are 


RANGE IN YARDS 

0 1000 2000 3000 4000 5000 



Figure 24. Average transmission anomalies, above 
and below thermocline. 


• 

AVERAG 

STRAIGH 

THEORE' 

OBSERVE 

T LINE FIT 
GCAL CUR 

D DIFFERS 
TED TO PC 
VE 

NCE 

INTS 







X 

// 




✓ 

_ 

/X 





0 1000 2000 3000 4000 5000 


RANGE IN YARDS 

Figure 25. Difference in transmission anomaly above 
and below thermocline. 

given later in this section. Since these results are 
subject to some uncertainty, it is useful to obtain an 
average value for this difference in transmission be¬ 
tween a shallow hydrophone in isothermal water and 
a deep hydrophone below the layer. Such an average 
has been obtained by averaging together all UCDWR 
runs made off San Diego under these conditions. 

The average curves found at 24 kc are shown in 
Figure 24. These curves include all rims in which the 
water was isothermal to more than 40 ft. The 
probable error of each curve, determined from the 


quartile deviation of the individual points, divided 
by the square root of the number of runs, is about 
1 db. The increased anomaly at short ranges for the 
deep hydrophone results from the vertical directivity 
of the sound projector. The difference between these 
two curves is plotted as a function of range in Figure 
25. It is evident that this difference, in decibels, in¬ 
creases linearly with range. The dotted line at less 
than 1,000 yd indicates the difference in anomaly 
that would presumably be found fora nondireetional 
projector. The dashed line represents the semi- 
empirical formula (13) discussed below. 

5.3.4 Studies of Layer Effect 
at 24 kc 

The average effect is large and significant. The 
way in which thermoclines of different depth and 
sharpness weaken the sound intensity below them 
is a detailed problem of both scientific and practical 
interest. First, the theoretical expressions for sound 
intensity below a thermocline are discussed and ap¬ 
plied to the average observational results. Secondly, 
the effect which thermocline depth and sharpness 
has on the measured sound intensity at depth is 
discussed in detail. Some early UCDWR studies 
are reported which fail to show the expected effects; 
a more detailed study is then given which indicates 
that general theoretical expectations are, in fact, 
fulfilled. 

Theory 

One might expect that at least in some cases the 
theory developed in Chapter 3 could be used to pre¬ 
dict the sound intensity below a layer of sharp tem¬ 
perature gradient. This expectation is supported by 
the discussion in Section 9.2.2, which shows that the 
intensities of explosive pulses agree rather well with 
the intensity calculations based on the ray theory. 
It is evident from Figure 23, on comparison of the 
solid line drawn through the circles with the dashed 
theoretical curve, that at ranges less than 2,000 yd, 
layer effect can in fact be explained on the basis of the 
simple ray theory. The predicted decrease of intensity 
results from the increased divergence of rays, which 
are bent sharply downward on passing through a 
temperature gradient. This increase of divergence is 
shown in the idealized diagram in Figure 2G. The rays 
are close together in the ideal isovelocity layer, and 
the intensity is therefore high; but below the layer 
of sharp gradient they are much further apart, re¬ 
sulting in much reduced intensity. 

























TRANSMISSION THROUGH A THERMOCLINE 


113 


TEMPERATURE 



Layer effect was examined theoretically, from the 
standpoint of ray acoustics, under “Combination of 
Linear Gradients’ in Section 3.4.2. The approximate 
formula obtained in Chapter 3 was 

- 4 = 10log (^)' < u) 

In this formula, A is the transmission anomaly, hi is 
the vertical distance between sound source and top 
of the thermocline, 0 h is the angle of inclination of the 
sound path at the hydrophone, R the range, and 0 O is 
the angle of inclination of the sound path at the sound 
source, as in Figure 27. To facilitate comparison with 
experiment, this expression can be further simplified. 
If the upward bending caused by the increasing pres¬ 
sure is unimportant, which will be the case at ranges 
which are not too long, then to an adequate approxi¬ 
mation, do may be replaced by hi/R. If d h is calculated 
by means of equation (81) of Chapter 3, equation (11) 
then becomes 


A = 10 log 



In most situations, the thermocline is not confined 
to a thin layer but is a hundred feet or more in thick¬ 
ness. Extensive intensity computations for simple 
types of bathythermograph slides are included in a 
report by WHOI. 31 Some of these have been repro¬ 
duced in Figure 25 of Chapter 3. It is evident from a 
study of these figures that most of the drop in inten¬ 
sity takes place in the top part of the gradient. With 
increasing hydrophone depth in the thermocline, the 
increase of total temperature change begins to be off¬ 
set by the larger inclination with which the sound ray 
enters the thermocline on its way to the hydrophone. 
More detailed theoretical calculations show that the 
temperature gradient in approximately the top D/3 ft 
of the thermocline should be important, where D is 


SOURCE 



the depth to the top of the thermocline, sometimes 
called layer depth. Since the exact choice of depth 
interval should not be very critical, it has been cus¬ 
tomary to use the temperature change AT in the top 
30 ft of the temperature gradient in computations of 
the intensity change to be expected theoretically. 

When there are several temperature gradients 
present, or when the sharpness of the gradient in¬ 
creases with depth, the theoretical intensity depends 
in a complicated way on the temperature pattern. 
However, an empirical study of the numerical in¬ 
tensity computations summarized in reference 31 
shows that the following procedure usually gives a 
moderately good approximation to the theoretical 
intensity found by ray tracing methods. Consider 
separately each 30-ft interval of the thermocline. 
Take the largest value of A found from equation (12); 
this then gives the theoretical intensity for a given 
initial ray inclination 6 when the ray reaches a depth 
about 4/3 of thedepthto the top of this 30-ft interval. 
This computed intensity also applies in theory to 
somewhat greater depths, since especially for deep 
thermoclines the intensity increases only relatively 
slowly with increasing depth below the depth of 
minimum intensity. 

Only that part of attenuation which is due to the 
sharp refraction at the top of the thermocline is taken 
into account in the expression (12) for the transmis¬ 
sion anomaly to points below the layer. If we assume 
that the attenuation due to absorption is 4 db per 
kyd, which is a reasonable estimate for 24-kc sound 
in an isothermal layer, the formula (12) is replaced 
by the following more realistic formula 

( 2AcR 2 \ 

A = 0.004 R + 5 log (^1 + J • (13) 

In formula (13), R is the horizontal range to the point 
where A is measured, Ac is the temperature decrease 
in the top 30 ft of the thermocline, and hi is the height 
of the sound projector above the top of the thermo- 


/ 1 + 


2A cR 2 

Coh\ 















114 


DEEP-WATER TRANSMISSION 


cline. The first term on the right in formula (13) 
represents the attenuation caused by absorption, and 
the second term represents the attenuation due to 
refraction. 

For the data plotted in Figures 24 and 25, the 
average depth to the top of the thermocline is about 
150 ft, which gives a value of 135 ft for hi. The aver¬ 
age value of Ac is about 15 ft per sec, which for a sur¬ 
face temperature of 70 F corresponds to a tempera¬ 
ture decrease of 3 F in the top 30 ft of the thermo¬ 
cline. If these numerical values are substituted into 
equation (13), and this term plotted against the 
range R, the dashed curve of Figure 25 results. The 
agreement between theory and observation is fairly 
close at ranges between 1,000 and 4,000 yd. 

This agreement is rather surprising since equation 
(13) is theoretically not valid at ranges so large that 
the upward bending in the isothermal layer becomes 
important and 6 h in equation (11) is no longer equal 
to hi/R. A more detailed theory, which takes into ac¬ 
count this upward refraction and assumes reflection 
of sound from a flat surface, would predict a shadow 
boundary at about 3,000 yd, with a very large anom¬ 
aly at greater ranges. Possibly, sound reflected from 
the irregular ocean surface, temperature microstruc¬ 
ture, or small systematic negative gradients near the 
surface, discussed in Section 5.4.2, might explain why 
equation (13) agrees so well with the facts beyond its 
expected range of validity. Regardless of the explana¬ 
tion, however, equation (13) may be regarded tenta¬ 
tively as a semi-empirical formula which may be 
used to predict the sound intensity below a ther¬ 
mocline of given depth and sharpness. A detailed 
comparison between this equation and the observa¬ 
tional data is given in Section 5.3.4. 

Early Studies 

One would expect from equation (13) that the dif¬ 
ference in intensity above and below the thermocline 
would depend both on the depth of the layer and on 
the magnitude of the temperature change in the 
thermocline. Two early studies along this line were 
made at UCDWR. Although these studies have not 
been conclusive and are largely superseded by the 
more recent results in the following section, they are 
given here for completeness. 

A preliminary plot of the intensity difference 
above and below the thermocline was made in a 
UCDWR internal report, 32 using data obtained in 
10 vertical runs, during which the hydrophone depth 
was slowly changed. A least squares solution, with A 


as the dependent variable and log (1 -f- 2Ac/f 2 /c«/i 2 ) 
as the independent variable, gave the relation 

/ 2AcR 2 \ , , 
A = —0.3 + 2.85 log ^1 -|—J (14) 

with h set equal to the thermocline depth, and c to the 
total velocity change from the isothermal layer to the 
measuring hydrophone. The measurements extended 
over a spread of 10 to 5,000 for 2AcR-/c 0 k 2 . Use of the 
velocity change in the top 30 ft of the thermocline 
would probably not have changed the results ap¬ 
preciably because of this large spread in 2AcR 2 /coh 2 . 
Thus these data indicated a difference of transmission 
anomaly only about half of that predicted by equa¬ 
tion (13); also, the scatter from the mean curve was 
very great. However, the temperature gradients near 
the surface were not specified, and an analysis of runs 
with isothermal water at the surface might be ex¬ 
pected to give better agreement. Of possible im¬ 
portance also is the fact that during the appreciable 
time required for vertical runs the temperature pat¬ 
tern could change appreciably. 

More recent analyses have dealt not with trans¬ 
mission anomalies but with values of Rao, the range 
at which the sound intensity is 40 db below the in¬ 
tensity measured with the hydrophone at 16-ft depth 
at a range of 100 yd. Further, a single parameter is 
frequently useful to characterize each transmission 
run, especially when a preliminary analysis of many 
runs is being attempted. For these reasons Rv> has 
been widely used in analyses of transmission data. 

Studies have been made of A R w , the difference in 
the R$q values determined above and below the layer. 
Since the values of A/? 4 o are based on differences be¬ 
tween intensities measured simultaneously in the 
isothermal layer and below the thermocline, it was 
hoped that these values would be less influenced by 
variability than individual values of Rw However, 
the study of A R w , given in a UCDWR internal re¬ 
port, 33 has yielded relatively few results, apart from 
providing general confirmation of the presence of 
layer effect. For isothermal layers deeper than 40 ft, 
the average value of A R w was 800 yd at 24 kc. How¬ 
ever, no correlation of A R i0 at 24 kc could be found 
with the depth or sharpness of the thermocline under¬ 
lying the isothermal layer, or with any other feature 
of the temperature distribution. 

This result may be attributed in part to the fact 
that /? 4 o above the thermocline apparently shows 
some correlation with both the depth and the sharp¬ 
ness of the thermocline. The data in reference 32 sug- 




TRANSMISSION THROUGH A THERMOCLINE 


115 



AT IN TOP 30 FEET OF THERMOCLINE IN OEGREES F 

Figure 28. Correlation between R i0 and sharpness of thermocline. 


gest that this variation is sufficiently similar to the 
variation of Rao below the thermocline that AR in 
would not be expected to show much predictable 
change with hydrographic conditions. However, the 
data analyzed in reference 32 are not sufficiently 
complete to allow definite conclusions. 

The large scatter of the observed data may also 
contribute to the failure to find any significant cor¬ 
relations in the study of AR W - Only half the observed 
values of A R i0 at 24 kc lay between 450 and 1,150 yd, 
corresponding to a quartile deviation of 350 yd. This 
is about what would be expected from an observa¬ 
tional error of 2 db in the measured transmission 
losses. In the isothermal layer, the value of R in is 
changed about 300 yd by a 2-db change in transmis¬ 
sion loss. Below the thermocline, the same change in 
transmission loss produces a change of only 175 yd in 
R i0 ; because of the steeper slope of the transmission- 
loss curve below the thermocline a smaller change of 
range is required to offset a change of transmission 
loss than in the isothermal layer. The square root of 
the sum of the squares of these two quantities is about 
350 yd, in agreement with the observed quartile 
deviation. This close agreement is somewhat sur¬ 
prising since some of the observed scatter is presuma¬ 
bly due to variations in hydrographic conditions. In 


any case, since the values of R 4 o in the isothermal 
layer introduce so much scattering in the values of 
A7? 40 , it is reasonable to expect that the analysis of 
A/hm is not an appropriate method for investigating 
the way in which Rao below the thermocline depends 
on detailed oceanographic conditions. 

Correlation with Depth and Sharpness of 
Thermocline 

Examination of the values of Rao below the ther¬ 
mocline shows that these are in fact correlated with 
both the sharpness and the depth of the thermocline. 
First, the data will be presented on the change of Rao 
with thermocline sharpness. All the values of R 4 o ob¬ 
tained by UCDWR when the depth to the top of the 
thermocline was between 100 and 200 ft and the 
hydrophone was below the thermocline are plotted 
in Figure 28 for different intervals of AT, the temper¬ 
ature change in the top 30 ft of the layer. The values 
of AT shown are only approximate, as a result of the 
grouping of the recorded data into four different 
groups, as follows: AT less than 0.7 F; AT between 
0.7 and 1.5 F; AT between 1.6 and 4.0 F; and AT be¬ 
tween 4.1 and 12.5 F. 

The crosses represent runs in which the hydro¬ 
phone was 100 ft or less below the top of the ther- 




























116 


DEEP-WATER TRANSMISSION 


moeline while for the circles the hydrophone depth 
was more than 100 ft below the top. From intensity 
contour diagrams like those shown in Figure 25 of 
Chapter 3, it is evident that for thermoclines within 
less than 100 ft from the surface, the computed in¬ 
tensity increases appreciably as the hydrophone goes 
from just below the layer to considerably greater 
depths. No simple formula has been derived for the 
increase of intensity in this case. The points plotted 
in Figure 28 indicate that for these deeper thermo¬ 
clines also, the value of R w tends to increase some¬ 
what with increasing depth of hydrophone below the 
top of the thermocline, provided the value of A7 1 is 
less than about 2 degrees. This result is also in gen¬ 
eral accordance with the predictions of intensity-con¬ 
tour diagrams. 


3500 


(/) 3000 

o 

<r 


uj 2500 
z 

o 

o 

2 

x 2000 
>- 

* 

o 

_i 

“ 1500 

a. 

o 


a 1000 


500 

50 100 150 200 250 300 

DEPTH TO TOP OF THERMOCLINE IN FEET 

Figure 29. Correlation between Ru, and depth to ther¬ 
mocline. Temperature difference in top 30 feet of 
thermocline, 1.6 degrees to 4.0 degrees. 


o 





0 

o 

X 

o 

X 

X X 

X 


c? 

oo 

X < 

o 

X < 

> oO 

U— 

riTx x 

% * 


X xx x 

WO KX 

X * 

< 

o 00 
o 

x x * 
*x 


o° cJ 

X 

c 

HYDROP 

BELOW 

HYDROP 

BELOW 

— THEORE 

USING £ 

o 

HONE 100 F 

TOP OF TH 

HONE MORE 

TOP OF TF 

TICAL CUR\ 

kT EQUAL 

EET OR LE 

ERMOCLINE 

THAN 100 

ERMOCLINE 

E FROM E 

TO 2.5° 

SS 

FEET 

QUATION 13 


The curve in Figure 28 is computed directly from 
equation (13), with a thermocline depth of 150 ft and 
a surface temperature of 70 F. It is evident from 
Figure 28 that the change of R w with changing tem¬ 
perature difference is, if anything, somewhat greater 
than can be explained on the basis of equation (13). 
This result is the reverse of that found in the empiri¬ 
cal equation (14). The many different points plotted 
in Figure 28 are not all completely independent since 
many were taken on the same day. Thus, the sam¬ 
pling error may be larger than might be expected from 


the number of points plotted. However, the data 
shown in Figure 28 are more extensive than those 
used in reference 32, and the result should therefore 
be more reliable. 

Similar data may be used to show the dependence 
of Ra o on thermocline depth. Values of Rio obtained 
with hydrophones below the thermocline, and with 
temperature differences of 1.6 to 4.0 F in the top 30 ft 
of the thermocline are shown in Figure 29. The circles 
and crosses have the same meaning as before. It is 
apparent that, for the shallower layers, the increase 
in intensity at depths well below the thermocline can 
become quite marked; this is in accordance with the 
theoretical expectations schematically presented in 
the intensity-contour diagrams of reference 31. 

The curve in Figure 29 shows theoretical values, 
computed from equation (13), with a velocity differ¬ 
ence Ac of 12 ft per sec, corresponding to a tempera¬ 
ture change of 2.5 F at a surface temperature of 70 F. 
The change in the median R w is approximately that 
predicted by equation (13). The quartile deviation, 
however, is of the same order of magnitude as the 
increase in median R w when the depth of the isother¬ 
mal layer is increased from 70 ft to 200 ft. 

The general trend in Figures 28 and 29 seems to 
indicate that equation (13) gives a rough approxima¬ 
tion to the median observed transmission. Though 
the spread is large, the data are not in disagreement 
with the predictions of that equation about the effect 
of changes in layer depth and thermocline sharpness. 
Thus equation (13) gives a moderately good fit to 
UCDWR transmission data. 

Additional data are required, of course, for more 
conclusive results. In particular, the number of varia¬ 
bles that might enter the problem is so great that other 
factors may be responsible for the apparent agree¬ 
ment between observations and the simple theory. 
Nevertheless Figures 28 and 29 indicate that equa¬ 
tion (13) provides a moderately satisfactory empirical 
fit for the present data. 

Some of these same results have been obtained in 
greater detail in an analysis of average transmission 
anomaly curves. This analysis 14 classifies the data 
according to the temperature code discussed in Sec¬ 
tion 5.1.3. The average anomaly curves for hydro¬ 
phones in or below the thermocline with d 2 equal to 
4 and to either 5 or 6 are given in Figures 30 and 31, 
respectively. These correspond to isothermal layers 
between 40 and 80 ft thick, and between 80 and 320 ft 
thick, respectively. In Figure 30, curve III, with the 
hydrophone between 20 and 160 ft below the top of 














TRANSMISSION THROUGH A THERMOCLINE 


117 



Figure 30. Average transmission anomalies for iso¬ 
thermal layer 40 to 80 feet thick. 



RANGE IN YARDS 

Figure 31. Average transmission anomalies for iso¬ 
thermal layer more than 80 feet thick. 


the thermocline, is significantly lower than curves IV 
and V, with the hydrophone between 120 and 360 ft 
below the top. Curve II apparently combines some 
runs with the hydrophone above the layer with 
others below the layer and is thus intermediate be¬ 
tween curve I (hydrophone in the isothermal layer) 
and curve III (hydrophone below top of thermocline). 
In Figure 31, curves III, IV, and V agree with each 
other, and apparently represent an average anomaly 
curve for a hydrophone below the thermocline. In 
agreement with equation (13), the anomalies are not 
so great as those found for a hydrophone just below 
a shallow layer. An analysis relating the average trans¬ 
mission anomaly below a thermocline to changes in 
the depth and sharpness of the thermocline might 
give more useful information than can be obtained 
with the temperature code used in Figures 30 and 31. 

5.3.5 Transmission at 60 kc 

The only other frequency for which data are avail¬ 
able on transmission below the thermocline is 60 kc. 


\ji 

_J 

UJ 

CD 

6 

UJ 

o 

z 

>- 

J 

< 

o 

z 

< 


o 

CO 

if) 

2 

if) 

z 

< 

tr 


RANGE IN YAR0S 



Figure 32. Average transmission anomalies at 60 kc, 
above and below thermocline. 



5 RANGE IN YAR0S 


Figure 33. Difference in transmission anomalies 
above and below thermocline. 

The average transmission anomalies both in the 
isothermal water above the thermocline and in the 
thermocline are shown in Figure 32. The difference 
between these curves at 1,000 and 2,000 yd is plotted 
in Figure 33; for comparison, the corresponding dif¬ 
ferences at 24 kc are also shown, which were taken 
from Figure 24. On the ray theory, these two curves 
should be identical if 0 o is the same for the two fre¬ 
quencies, since the increased divergence resulting 
from downward bending should be independent of 
frequency. The agreement between the 24-kc crosses 
and the 60-kc circles in Figure 33 is not too close; 
however, such a comparison of data cannot be reliable 
unless the measurements were made under similar 
thermal conditions. 













































118 


DEEP-WATER TRANSMISSION 





4880 4930 4980 

SOUNO VELOCITY IN FT PER SEP 


DATE 7-6-1943 
TIME _ 

BT CLASS NAN 
WATER DEPTH 615 FM 

SEA _ 

SWELL_ 

WIND ___ 


Figure 34. Sample transmission anomaly plot. 


5.4 TRANSMISSION WITH NEGATIVE 
GRADIENTS NEAR SURFACE 

In some regions at certain times, negative tem¬ 
perature gradients in the upper 50 to 100 ft of the 
ocean are very common. Especially in coastal waters 
and during the summer months the temperature dif¬ 
ference between the surface and 30 ft frequently ex¬ 
ceeds 0.3 F. Temperature patterns of this type are 
usually highly variable. 

Many of the transmission measurements at 
UCDWR were made with negative gradients at the 
surface. The present section discusses these data. 
While emphasis is placed on temperature patterns 
for which the temperature difference from the surface 
to 30 ft is 0.4 F or more (NAN and CHARLIE pat¬ 
terns), some discussion is included of those cases where 
the top 30 ft is isothermal but the top of the ther- 
mocline is less than 100 ft below the surface. A more 
rigid division of the acoustic data by the different 
types of temperature-depth patterns is not possible, 
since the different analyses that have been made 
have used somewhat different classifications. 

In the following pages, a discussion will first be 


given of the general types of transmission anomalies 
that are found with temperature gradients near the 
surface. Subsequently, more detailed discussions will 
be given, first for those situations where the temper¬ 
ature gradient in the top 30 ft is large, about 0.7 F or 
more, and second, for smaller gradients in the top 
30 ft. 

Because of an almost complete lack of analyzed 
data at other frequencies, this discussion is largely 
confined to results obtained at 24 kc. A few results 
obtained at 60 kc are described at the end of this 
section. The bulk of the results come from two re¬ 
ports by UCDWR, one issued in 1943' 11 and the other 
in 1945. 14 Use is also made of individual transmission 
anomaly curves obtained from UCDWR which have 
not been published. 

An examination of the transmission anomalies 
measured with temperature gradients near the sur¬ 
face shows that most of the observational curves may 
be divided into two types. In the first of these, the 
anomaly does not change rapidly with range at very 
close ranges or at very long ranges, but drops very 
rapidly at a range somewhere between 500 and 2,000 
yd. In the second type, the transmission anomaly in- 








































TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


119 


(D 

O 


> 

l 

< 

2 

O 

z 

< 

z 

o 

v> 

2 

z 

< 

o: 

t- 


0 


50 


100 


150 


RAY DIAGRAM 




4080 4930 4980 



RANGE IN YARDS 

Figure 35. Sample transmission anomaly plot. 


creases linearly with range out to long ranges. Sample 
types of these two curves are shown in Figures 34 and 
35, respectively. With each figure is included a 
smoothed temperature-depth plot and a correspond¬ 
ing ray diagram. 

The relative number of straight-line anomaly 
curves for different hydrographic conditions has been 
investigated. 13 The number of straight-line graphs 
was found for different values of the temperature dif¬ 
ference from 0 to 30 ft and for different values of the 
computed limiting range at the hydrophone depth 
used. The results of this study are given in Table 1. 
The accuracy with which observed curves could be 
fitted by straight lines has already been discussed in 
Section 5.2 where the high probability of straight-line 
graphs in isothermal water was also pointed out. 

The classification of bat hy thermograms into 
MIKE, CHARLIE, and NAN patterns on the basis 
of the temperature difference from 0 to 30 ft has al¬ 
ready been discussed in Section 5.1.4. These patterns 
are indicated in Table 1; the dividing line between 
NAN and CHARLIE patterns, usually defined as 
1/100 of the surface temperature, is about 0.7 F for 


Table 1. Relative number of straight-line anomaly 
graphs. 


Temperature 

difference 

0 to 30 ft in °F 

Number of 
straight 
graphs 

Number of 
graphs not 
straight 

Tempera¬ 

ture 

pattern 

0.0 

40 

0 

MIKE 

0.1 

25 

0 

MIKE 

0.2 

18 

2 

MIKE 

0.3 

5 

i 

MIKE 

0.4 

8 

7 

CHARLIE 

0.5 

4 

1 

CHARLIE 

0.6 

3 

7 

CHARLIE 

0.7 or more 

8 

42 

NAN 

Total 

111 

60 


Range to shadow 
boundary at hy¬ 
drophone depth 
in yards 

0-500 

0 

8 


500-1,000 

10 

22 


1,000-1,500 

5 

16 


1,500-2,000 

12 

6 


2,000-3,000 

15 

5 


3,000-4,000 

31 

i 


4,000 or more 

38 

2 


Total 

111 

60 






















































120 


DEEP-WATER TRANSMISSION 


RAY DIAGRAM 




4890 4940 4990 

SOUND VELOCITY IN FT PER SECOND 



DATE 4-16- 

94 4 

TIME 1650 


BT CLASS 

NAN 

WATER DEPTH 

680 FM 

SEA 1 


SWELL 3 


WIND FORCE 2 


RANGE IN tARDS 


Figure 36. Sample transmission anomaly plot. 


most of the runs made off San Diego. While these pat¬ 
terns are by no means fundamental, Table 1 suggests 
that they provide a natural classification for the 
data. 

It is evident from these tables that when sharp 
temperature gradients are present at the surface, 
straight-line graphs are unlikely. In the following 
discussion, transmission runs made with sharp tem¬ 
perature gradients present in the top 30 ft are there¬ 
fore treated separately from those made in the pres¬ 
ence of weaker gradients. 

5.4.1 Sharp Temperature Gradients 
in Top 30 ft 

When the temperature gradient in the top 30 ft is 
sharp and extends all the way to the surface, theory 
predicts a sharp downward bending of the sound 
beam, and a shadow zone of low sound intensity at 
fairly close ranges. This expectation is generally ful¬ 


filled, provided that the difference of temperature be¬ 
tween the surface and 30 ft is more than 0.7 F, and 
the hydrophone is above 100 ft. Samples of such 
curves are shown for the shallow hydrophones in 
Figures 36 and 37 as well as in Figure 34. Since the 
surface temperature for most of the UCDWR trans¬ 
mission runs off San Diego was in the neighborhood 
of 70 F, this critical value is 1/100 of the surface 
temperature, which is the dividing line between 
NAN and CHARLIE patterns. 

While the ray theory can explain the qualitative 
features of the sound field observed with NAN pat¬ 
terns, it cannot predict exactly the transmission 
anomalies to be expected under different conditions. 
The detailed dependence of sound transmission under 
these conditions on the temperature structure and on 
the hydrophone depth is an important subject on 
which considerable data are available. This informa¬ 
tion is summarized in the following pages. First, the 
correlation between transmission data and the com- 




































TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


121 




SOUND FIELD DATA 


50 60 70 

TEMPERATURE IN DEGREES F 



Figure 37. Sample transmission anomaly plot; 


puted range to the shadow boundary is discussed. 
This is followed by a more detailed statistical investi¬ 
gation of average transmission anomaly curves for 
different types of temperature patterns and for dif¬ 
ferent hydrophone depths. Finally, separate discus¬ 
sions are given of the observed slope of the transmis¬ 
sion anomaly near the shadow boundary and of the 
sound observed in computed shadow zones. 

Computed Range to the Shadow Boundary 

When the temperature gradient extends right up 
to the surface, and a shadow zone is predicted from 
the simple theory, the range to the break in the ob¬ 
served transmission anomaly curve might be ex¬ 
pected to equal the computed range to the shadow 
boundary. More frequently, however, the range to 
the break is less than this value, as shown in Figures 
34, 36, and 37. A detailed comparison between the 
ray diagram and the transmission anomaly curves is 
complicated by the variability of temperature condi¬ 
tions, already discussed in Section 5.1.3. However, a 


brief analysis of some of the data has been carried 
out, which averages the range to the shadow bound¬ 
ary computed from bathythermograms taken at the 
beginning and end of the transmission run. 

The resulting plot of these data is shown in Figure 
38, taken from reference 34. This figure includes data 
taken during two days of measurements. The dashed 
line has been fitted visually to the observed points. 
The correlation between the observed and the theo¬ 
retical values is moderately good, considering the 
variability of the temperature structure. However, 
the observed ranges to the shadow boundary seem 
to be systematically less than the predicted values, an 
effect which is difficult to attribute to changes in 
ocean temperature or errors in reading the bathy¬ 
thermograph slide. The data presented in Section 
9.2.3 on the change in shape of the pulse in the shadow 
zone seem to indicate that the range to the break in 
the transmission anomaly curve is correctly identified 
as the boundary of the shadow zone. Thus, Figure 38 
may present a real discrepancy, although the amount 





























122 


DEEP-WATER TRANSMISSION 



COMPUTED RANGE IN YARDS TO SHADOW BOUNDARY 


Figure 38. Correlation between break in transmission 
anomaly plot and computed range to shadow boundary. 



COMPUTED RANGE IN YARDS TO SHADOW BOUNDARY 

Figure 39. Correlation between h\„ and computed 
range to shadow boundary. 


of data is too limited to permit very definite con¬ 
clusions. Possibly the presence of water waves on the 
ocean surface lowers the effective level of the surface 
and brings the shadow boundary in to closer range 
than would be expected for a flat surface. 

A somewhat more practical quantity found from 
the transmission curves is the range R 40 at which the 
sound intensity is 40 db below the intensity at 100 yd. 
The significance of this quantity has already been 
discussed in Section 5.3.4. A plot of R w against com¬ 
puted limiting range is shown in Figure 39. The cor¬ 
relation is again only fair. The dashed curve of best 
fit was chosen visually, as in Figure 38. 

Figures 38 and 39 cannot be used to give reliable 
average results because of the paucity of data in¬ 
cluded. They indicate in a general way the correla¬ 
tion that may be expected between detailed computa¬ 
tions of limiting rays and observed transmitted sound 
intensities. 



Figure 40. Average transmission anomalies for NAN 
patterns, shallow hydrophone. 


Average Transmission Anomaly 

A statistical average of the measured transmission 
anomalies for different temperature conditions is 
given in reference 14, based on the temperature- 
depth code described in Section 5.1.4. The data for 
shallow hydrophones (1G to 30 ft) are discussed first. 
Corresponding data for deeper hydrophones are given 
in the next section. Three curves for shallow hydro¬ 
phones in water with sharp temperature gradients in 
the top 30 ft (NAN patterns) are given in Figure 40. 
Each curve represents an average for a different value 
of Do, the depth at which the temperature is 0.3 F less 
than the surface temperature. An examination of the 
bathythermograph data shows that for 1) 2 less than 
5 ft, the main thermocline in every case extended all 
the way to the surface, resulting in very sharp down¬ 
ward bending of the beam. For Do between 5 and 
20 ft, a layer of small gradient overlay the thermo¬ 
cline, which usually extended up to within 20 ft of the 
surface, while for Do between 20 and 30 ft the main 
thermocline was always deeper than 20 ft and usually 
between 20 and 30 ft from the surface. To indicate 
the scatter of the individual points making up these 
average curves, all anomalies for D 2 between 5 and 
20 ft are plotted in Figure 41. Half of the points lie 
within about 4 db of the mean curve. The scatter 
shows some tendency to increase with range and is 
significantly greater than that found with a hydro¬ 
phone in deep isothermal water (see Figure 15). There 
is no significant change either in the mean curve or 
in the scatter if values of D 2 between 5 and 10 ft and 
between 10 and 20 ft are considered separately. The 
curve for Z) 2 between 20 and 30 ft is based on rela- 
























TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


123 



tively few points, and the individual points show a 
large scatter. Thus, this curve is not very reliable. 

Average anomaly curves have also been computed 
for different values of D u the depth at which the 
temperature is 0.1 F less than the surface tempera¬ 
ture. For a fixed value of D 2 , these average curves 
show no systematic variation with changes in D i. 
With NAN patterns, gradients below 40 ft also have 
a negligible effect on the average transmission anoma¬ 
lies measured with a shallow hydrophone. 

The change of sound transmission with changing 
Zb is to be expected on theoretical grounds. It is evi¬ 
dent from Figure 35 that the range to the computed 
shadow boundary becomes much extended when a 
shallow layer of weak gradient overlies the major 
temperature decrease. When thermal microstructure 
or surface reflections are taken into account, sound 
in a thin surface layer of weak gradient can evidently 
be propagated out to substantial ranges, with some 
sound continually being bent down into the layer of 
sharp gradient. Thus no shadow zone is to be ex¬ 
pected in this situation, and straight-line graphs of 
the type shown in Figure 35 are likely. The high at¬ 
tenuation observed in this thin layer is also generally 
found in shallow isothermal layers, and is discussed 
again in Section 5.4.2. 


Dependence on Hydrophone Depth 

In accordance with expectations, the range to the 
observed shadow zone — either R i0 or the range to 
the break in the transmission anomaly curve — in¬ 
creases with increasing hydrophone depth. This 
effect, when present, results in a predicted increase 
of maximum echo range with increasing submarine 
depth; this increase was observed in early practical 
echo-ranging trials. 35,36 The same effect is shown 
clearly in Figures 36 and 37; the drop in intensity 
for the deep hydrophones is found at ranges sub¬ 
stantially larger than for the shallow hydrophone. 
When a deep isothermal layer is present below the 
sharp surface gradient, this effect would be expected 
to be very marked, on the basis of ray theory. A 
sample run in one of the rare observed situations of 
this type is shown in Figure 42, with the accompany¬ 
ing temperature-depth plot. 

The statistical analysis of the data reported in 
reference 14 provides an indication of the average 
change of intensity with depth. The average curves 
found for hydrophones at different depths below 50 ft 
are reproduced in Figure 43. The upper and lower 
plots correspond, respectively, to intervals of 0-10 
and 10-20 ft for Zb, the depth at which the tempera¬ 
ture is 0.3 degree less than the surface temperature 































124 


DEEP-WATER TRANSMISSION 



Figure 42. Sample transmission anomaly plot; NAN pattern with deep isothermal layer below. 


Insufficient data are available for NAN patterns with 
Do between 20 and 30 ft to yield much information on 
the change of transmission anomaly with hydrophone 
depth for that case. The scatter of the individual 
points averaged to yield Figure 43 is moderately 
large. Since only about ten runs were averaged to 
give each curve, these average curves have a probable 
error of between 2 and 3 db. 

Attenuation Coefficient at Shadow Boundary 
The mechanism by which sound penetrates the 
shadow zone is of theoretical and practical interest. 
Information about this mechanism may be obtained 
by comparing the rate of increase of transmission 
anomaly near the shadow boundary with that com¬ 
puted from the diffraction of sound. The slope of the 
transmission anomaly beyond the break is very high, 
usually between 20 and 70 db per kyd when the sharp 
temperature gradient extends all the way to the sur¬ 
face ( D -2 less than 10 ft). We shall call this slope a' and 


may regard it as a sort of “local attenuation coef¬ 
ficient,” that is, 


The local attenuation coefficient defined by equation 
(15) is not to be confused with the actual attenuation 
coefficient characterizing the transmission from the 
sound source to the range R, defined as l,000A//f. 

The observed slope beyond the break is to be com¬ 
pared with the local attenuation coefficient at the 
shadow boundary which would result from diffrac¬ 
tion. From Section 3.7, we have the following formula 
for o' in the case of a linear velocity gradient. 



In formula (16) c is the sound velocity in yards per 
second; dc/dy is the velocity gradient in feet per 
second per foot; and o' is in units of decibels per yard. 

















































TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


125 


If the temperature difference between 0 and 30 ft 
is about 1 F, and the surface temperature is about 
70 F, then for sound of 24,000 cycles equation (16) 
gives an attenuation of about 0.1 db per yd, or about 
100 db per kyd. This is at least twice as great as the 
values usually obtained. The discrepancy seems 
somewhat too great to be explained as observational 
error, although the fact that observed and predicted 
attenuations are of the same order of magnitude is 
suggestive. It is possible that the presence of thermal 
microstructure may explain this difference. No at¬ 
tempt has been made to correlate the observed at¬ 
tenuation across the shadow boundary with either 
the frequency/or the velocity gradient dc/cly at the 
surface. 


Scattered Sound in the Shadow Zone 

Most transmission anomaly plots for NAN pat¬ 
terns are characterized by a nearly constant trans¬ 
mission anomaly well beyond the computed boundary 
of the shadow zone, amounting to between 40 and 
60 db and extending out to the limit of measurement 
at about 5,000 yd. 

An examination of the oscillograph record shows 
that the signal received at ranges between 1,500 and 
about 3,000 yd bears very little relation to the shape 
or length of the original pulse. Figure 44 shows the 
signals received at different ranges when a marked 
negative gradient was present at the surface. At 
moderately short ranges, less than 1,000 yd, the re¬ 
ceived signal reproduces the outgoing pulse rather 
faithfully. At moderate ranges into the shadow zone, 
however, the signal has an appearance similar to that 
of reverberation, and is much prolonged, as shown in 
trace C made at 1,340 yd. At these intermediate 
ranges, the use of peak amplitudes in reading each 
signal gives a value about 7 db higher than the use of 
average intensities. For coherent 100-msec signals, 
the difference between peak amplitude and rms am¬ 
plitude is negligible. At slightly longer ranges even the 
few traces of the direct pulse, visible for some of the 
signals in trace C, completely disappear. At the long¬ 
est ranges the signal begins again to resemble the 
emitted pulse, as shown in trace D. 

The intensity of the sound received in the shadow 
zone increases with increasing pulse length. The 
observed difference in intensity for 100-msec and 
10-sec pulses is shown in Figure 45. In the shadow 
zone at intermediate range, where the received signal 
is prolonged and incoherent, the effect is large, 
amounting to between 4 and 8 db. At very long 


RANGE IN YARDS 



I 50^ h •< 100 FEET 

II 100^ h -200 FEET 

III 200S h — 300 FEET 

IV 300gh —400 FEET 



Figure 43. Average transmission anomalies for NAN 
patterns with hydrophone deeper than 50 feet. 


ranges and within the direct sound field at shorter 
ranges the signal is coherent and the difference is 
small. This small difference results from the fact that 
the peak amplitude of a long signal, which fluctuates 
from minimum to maximum values many times, 
tends to be greater than the peak amplitude of a 
short signal, which may be altered by fluctuation to 
a low value. 















126 


DEEP-WATER TRANSMISSION 


RADIO SOUND 
SIGNAL SIGNAL 


450 

YDS 


73.5 db 
ATTENUATION 


RADIO 

TRACE 


SOUND 

TRACE 


-=-j- 


RADIO SOUND 


A RECORDS AT 450 YARDS 



830 YDS 


43.5 db 
ATTENUATION 










L 

























- L 

*- 

- 



*- 




RADIO 


SOUND 


B RECORDS AT 830 YARDS 



1340 YDS 

- 

24 db 

ATTENUATION 






' 

U * 





’* ' . # 







-* 












RADIO 

SIGNAL 


DIRECT BOTTOM 
SOUND SIGNAL REFLECTION 


C RECORDS AT 1340 YARDS 


4350 YDS- 


9 db ATTENUATION 


D RECORDS AT 4350 YARDS 

HYDROPHONE DEPTH = 75 FEET OCEAN DEPTH = 615 FATHOMS 


Figure 44. Records of received signals at various ranges under downward refraction. 



RANGE IN YARDS 


Figure 45. Increase of average peak amplitudes for 
long pulses. 

The sound received in the shadow zone, at least out 
until about 2,500 or 3,000 yd, is probably sound 
scattered from the main beam at considerable depth 
up to the hydrophone near the surface. The scatter¬ 
ing coefficient required to explain the observations 
can be readily estimated from the transmission 


anomalies of the long pulses. The transmission anom¬ 
aly for 100-msec pulses in the shadow zone is usually 
between 40 and 60db. Figure 45 shows that about4to 
8 db must be subtracted from these values to find the 
transmission anomaly of the 10-sec pulse, which may 
be regarded as essentially a continuous tone in these 
observations. On the other hand, 7 db must be added 
to find the anomaly in terms of average intensity 
instead of average peak amplitude. Since these two 
corrections about cancel out, 40 to 60 db is the trans¬ 
mission anomaly for the intensity of long pulses in 
the shadow zone. 

A theoretical value for this transmission anomaly 
may be computed on a somewhat simplified picture. 
Although the scattered sound is itself refracted by 
the prevailing vertical velocity gradient, most of the 
scattered rays are inclined so steeply that they may 
be regarded as straight lines. With this assumption, 





































































































































TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


127 



the scattered sound is transmitted to the entire ocean 
as if refraction were not operating. Also, for the pur¬ 
pose of calculating the sound scattered into the 
shadow zone, the actual direct beam may be replaced 
by a tilted beam traveling in a straight line, as in 
Figure 46. 

To calculate the scattered sound received when a 
long (10-sec) pulse is sent out, it may be assumed 
that sound scattered from the entire length of 
the beam is received at the hydrophone. Let the total 
initial power in the beam be denoted by J. If attenua¬ 
tion is neglected, this energy will remain constant as 
the sound travels outward. If the scattering coef¬ 
ficient is m per yard, a fraction m of the sound energy 
will be scattered per yard of travel of the beam (see 
Chapter 2 of Part II). This energy will be scattered 
in all directions; and the intensity of the sound scat¬ 
tered from this cross section of the beam 1 yd thick 
and reaching the hydrophone at a distance r' yd away 
will be mJ/4-irr' 2 . While in actual fact the sound 
scattered from the lower side of the beam will be more 
weakened than that scattered from the upper side, 
the distance r' from the hydrophone to a point on the 
axis of the beam should be a reasonable approxima¬ 
tion for each separate cross section of the beam. Let l 
represent the distance from the projector to the point 
where scattering is taking place. The sound scattered 


between l and l + dl is thus (mJ/4irr' 2 )dl] and the 
total scattered sound received at the hydrophone is 

r ro mJ mJ T” dl 
s = Jo = Wo (L -l) 2 + d 2 (U) 

where the quantities L and d have the meanings 
shown in Figure 46. The integration yields approxi¬ 
mately 



While in the general case m will be a complicated 
function both of position in the ocean and of the 
direction in which the scattered sound is measured, 
here m is assumed to be constant. Equation (18) thus 
refers in practice to an average value of m. 

The expression (17) does not take into account the 
transmission anomaly resulting from absorption or 
refraction. When a sound beam is refracted sharply 
downward, the intensity in the direct sound field is 
not reduced much below the inverse square value. 
The scattered sound, which reaches the hydrophone 
at steep angles, is also relatively unaffected by re¬ 
fraction. The absorption must be considered, how¬ 
ever. For points in the sound beam to the left of the 
point B in Figure 46, the sum of the absorption loss 
for direct and scattered sound will not depend much 









128 


DEEP-WATER TRANSMISSION 


on whether the sound was scattered close to the pro¬ 
jector or some distance out. Sound scattered from 
points to the right of the point B will suffer a two- 
way absorption loss, and may be neglected. There¬ 
fore, to calculate the sound intensity at the hydro¬ 
phone, taking absorption into account, we must first 
integrate equation (17) with the infinite upper limit 
replaced by L, obtaining approximately mJ/Sd; then 
we must multiply this value by some factor to take 
the absorption into account. Because we are neglect¬ 
ing the sound scattered to the right of B, we may con¬ 
sider that the sound reaching the hydrophone has 
traveled a total path length, on the average, equal to 
R, the range from the projector to the hydrophone. 
If a is the attenuation coefficient in decibels per yard, 
the intensity I s is therefore given by 

I s = ^10-° B/1 °. (19) 

8 d 

Equation (19) was derived without considering 
the possibility that sound could be scattered up to 
the surface by points to the left of B and reflected 
back to the hydrophone. We can allow for this extra 
intensity due to surface reflection by multiplying the 
expression (19) by 2 . Our final result is 

Is = -y-ylO _aff/1 °. (20) 

4a 

It is convenient to restate equation (20) in decibels: 
10 log I s = 10 log m -f 10 log J — 10 log (4 d) — aR. 

( 21 ) 

The total emitted power J is related to F, the 
power output per unit solid angle in the direction of 
the projector axis, by the formula 

10 log J = 10 log (4 t wF) + D, (22) 

where D is the directivity index of the projector. 
Combining equations (22) and (21) gives 

10 log I s = 10 log m + 10 log F + D — 10 log d 

— aR + 10 log 7 r. (23) 

The transmission anomaly A of the scattered radia¬ 
tion is given by 

A = 10 log F — 20 log R — 10 log I s . 

If R sin 9 is substituted for d, from Figure 46, and I s 
is taken from equation (23), the transmission anom¬ 
aly A becomes 

A = — 10 log 72 — 10 log m — D + aR — 10 log x 

+ 10 log (sin 9). (24) 


For a total temperature decrease of about 20 de¬ 
grees in the thermocline, the limiting ray PSC in 
Figure 46 bends downward below the thermocline at 
an angle of 12 degrees. Thus, a typical value of 6 is 
12 degrees. For a directional transducer of the type 
normally used in echo ranging, the directivity index 
D is —23 db. If an absorption coefficient a of 4 db per 
kycl is used in equation (24), and D and 9 are set 
equal to —23 db and 12 degrees, respectively, then 
A is nearly constant from 1,000 to 3,000 yd, and 
equals —10 log m — 14. Since A is observed to lie 
between 40 and 60 for transmission to points well in¬ 
side the shadow zone, 10 log m is between —54 and 
— 74. Because the receiver is directional in a vertical 
plane, these values for 10 log m must be increased 
somewhat to take account of the directivity pattern 
of the receiver. An examination of the receiver pat¬ 
terns in reference 34 indicates that this correction 
should be about 6 db. Thus, we finally have for 
10 log m a value between —48 and —68 db. 

This result is in general agreement with the value 
of — 60 + 10 db for the scattering coefficient of 
volume reverberation given in Chapter 4 of Part II. 
A value greater than —40 db seems definitely ruled 
out by the observations. Thus one may conclude that 
the scattering coefficient for sound at angles between 
roughly 10 and 120 degrees is not more than about 
10 db greater than for the backward scattering which 
gives rise to reverberation. It is possible that the 
scattering of sound by the volume of the sea is the 
same in all directions. More exact conclusions would 
require simultaneous determinations of reverberation 
and sound scattered into the shadow zone. In addi¬ 
tion, the change of scattering coefficient with depth, 
frequently observed in the deep scattering layers dis¬ 
cussed in Chapter 14, would demand consideration. 
The present very rough analysis is adequate, how¬ 
ever, to indicate that the attenuation observed in 
deep isothermal water is not the result of scattering, 
unless one makes the improbable hypothesis that 
scattering in the isothermal layer is very much greater 
than the scattering in the thermocline. If an attenua¬ 
tion coefficient of 4 db per kycl or 4 X 10 _:, db per yd is 
attributed entirely to scattering, the scattering coef¬ 
ficient m would be 10 logi 0 e times a or 1.7 X 10~ 2 , 
giving more than —20 db for 10 log m. This is 20 db 
greater than the maximum possible value of m con¬ 
sistent with the low intensity of sound observed in 
the shadow zone. If not all the sound in the shadow 
zone is due to scattering, the disparity becomes even 
greater. 



TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


129 


i 



0 100 50 33 25 20 


DEPTH D t TO THERMOCLINE IN FEET 

Figure 47. Attenuation coefficient above the thermocline. 


Although the scattered sound observed at ranges 
between 1,000 and 2,500 yd is readily explained, the 
coherent signals received in the shadow zone at 4,000 
yd are less simply explained. These coherent signals 
could be produced by a suitable variation of the 
scattering coefficient m with depth similar to those 
found in deep scattering layers. A preliminary 
UCDWR analysis of reverberation measurements 
made with a vertical projector at the same time as 
transmission measurements indicates that this hy¬ 
pothesis is correct; the scattering coefficients found 
by these two methods agree to within a few decibels. 

5.4.2 Weak Temperature Gradients 
in Top 30 ft 

When the temperature gradients in the top 30 ft 
are intermediate—CHARLIE patterns—it has been 
clearly demonstrated in Table 1, that at least half of 
the transmission anomaly curves are approximately 
straight lines while the others have more complicated 
shapes. Thus the type of transmission likely to be en¬ 
countered is highly unpredictable. This observational 
result may be in part caused by the rapid variability 
of temperature conditions for this type of pattern; 
small changes of temperature, of the sort very com¬ 
mon near the surface, can change the theoretical ray 
diagram completely in a matter of minutes. Various 
methods have been developed for analyzing the trans¬ 


mission conditions to be expected with these patterns. 
Since the average transmission loss observed with 
shallow MIKE patterns is very similar to that ob¬ 
served for CHARLIE patterns, these two tempera¬ 
ture types are combined in the present discussion. 
Most of this section refers to average results obtained 
at 24 kc. Some special temperature distributions are 
discussed at the end of this section. 

Attenuation Coefficients 

In reference 13, an attenuation coefficient was 
found from the slope of each straight-line transmis¬ 
sion anomaly graph. Attempts to correlate these 
coefficients with various temperature differences 
either in the surface layers or in the thermocline were 
not very successful. However, a significant correla¬ 
tion was found with the depth D T to the thermocline. 
The plots showing this correlation are reproduced in 
Figures 47 and 48. A least-squares solution gave the 
following equations of best fit: 

Above the thermocline a = 3.5 4-- (25) 

D t 

260 

Below the thermocline a = 4.5 4-• (26) 

Dp 

Although these mean curves are unquestionably sig¬ 
nificant, only half of the individual points lie within 
2 db of (he values predicted from equations (25) and 
(26). 






















130 


DEEP-WATER TRANSMISSION 


» 



d t = oepth to thermocline in feet 

Figure 48. Attenuation coefficient below the thermocline. 


Figure 48 shows an increase of attenuation below 
the thermocline with decreasing thermocline depth. 
This effect has already been noted in Section 5.3.4 for 
thermocline depths greater than 40 ft; the change of 
Rao with layer depth (Figure 29) was shown to be 
consistent with the theoretical curve based on equa¬ 
tion (13). It is apparent from Figure 47 that this same 
effect persists to much shallower layers. A comparison 
of Figures 47 and 48 indicates that layer effect in¬ 
creases steadily with decreasing depth of the ther¬ 
mocline. This result is also consistent with expecta¬ 
tions based on equation (13). 

The increase of the attenuation coefficient in the 
isothermal layer as the depth of the layer decreases 
is quite marked in Figure 47. It may be noted that 
the values shown for D T between 20 and 30 ft are not 
inconsistent with the attenuation coefficient of about 
13 db per kyd found from the upper curve in Figure 
40, drawn for D 2 between 20 and 30 ft. The origin of 
this high attenuation is hard to explain. As sound 
travels along through the layer of nearly isothermal 
water, with the sound rays continually reflected from 
the surface and distorted by temperature microstruc- 
ture, it maybe expected that a certain fraction of the 
sound would be bent out of the isothermal layer in 
each yard of sound travel. Any such sound reaching 
the thermocline will be bent down so sharply that it is 
unlikely to return to the isothermal layer. It is possi¬ 
ble that a quantitative theory along these lines, based 


on more accurate information on the properties of the 
isothermal layer, may explain the observed decrease 
of attenuation with increasing thickness of the layer. 
In any case there is little question as to the reality 
of the effect noted in Figures 47 and 48. 

RANGE IN YARDS 



Figure 49. Average transmission anomalies for 
MIKE and CHARLIE patterns (hydrophone shallow). 


Whether the scatter evident in these figures is 
greater than can be explained by the observational 
scatter of all observed transmission anomalies is not 
evident from an examination of reference 13. It has 
already been noted, in Chapter 4, that for most of the 
UCDWR data transmission anomalies determined by 
averaging 5 successive received pings have a probable 
error of about 2 db as a result of hydrophone direc¬ 
tivity, training errors, and sampling errors; the errors 
























TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


131 


RANGE IN YARDS 



Figure 50. Individual anomalies for />. between 20 and 40 feet. 


due to calibration do not affect the accuracy of the 
attenuation coefficient determined by fitting a 
straight line to the observed anomalies. With such a 
scatter, the uncertainties in the attenuation coef¬ 
ficient a can be very substantial for those runs in 
which the length of run was short. However, it does 
not seem likely that observational error can account 
for all of the scatter shown in Figures 47 and 48. 

The attenuation coefficients shown in Figures 47 
and 48 should probably not be used for estimating 
transmitted sound intensities when temperature gra¬ 
dients are present in the top 30 ft. The results are 
valid on the average when the transmission anomaly 
graph is a straight line, but unfortunately there is no 


way of predicting whether or not the measured sound 
intensities will yield a straight line, except when the 
top 30 ft are isothermal. Exclusion of those situations 
where the transmission anomaly curve is not a 
straight line may be expected to give results system¬ 
atically different from those obtained when runs are 
classified oidy by the temperature distribution. 
Therefore the average anomaly curves given below 
are preferable as a tool for estimating the sound in¬ 
tensities to be expected in any situation. 

Average Transmission Anomalies 

The average transmission anomalies obtained with 
MIKE and CHARLIE patterns have been combined 




















132 


DEEP-WATER TRANSMISSION 


in reference 14 to give average curves. The curves for 
a shallow hydrophone (16 to 30 ft) are reproduced in 
Figure 49. The curves are again plotted for different 
values of Do. 

To illustrate the scatter of the individual observa¬ 
tions, all individual anomalies averaged to give the 
curve for Do between 20 and 40 ft are shown in Fig¬ 
ure 50. The open circles represent the anomalies for 
MIKE patterns, the solid circles those for CHARLIE 
patterns; no systematic difference is apparent be¬ 
tween these two sets of points. The upper and lower 
quartiles of the distribution are shown by dashed 
lines. The increase of spread with increasing range is 
very marked and is an evidence of the unpredicta¬ 
bility of transmission conditions for such shallow 
isothermal layers. The quartile spread at short range 
is much smaller and represents the more normal 
scatter, apparent also in Figures 15 and 41. 



Figure 51. Average transmission anomalies for D. 
between 20 and 40 feet. 


The change of transmission anomaly with changing 
hydrophone depth has not been analyzed separately 
for MIKE, NAN, and CHARLIE patterns. The 
average results for D 2 between 20 and 40 ft, combin¬ 
ing results for all three patterns, is shown in Figure 
51. The change with depth down to 200 ft is negligi¬ 
ble, but at greater depths an appreciable increase in 
sound intensity is noted. This change is greater than 
the probable error of each curve resulting from the 
internal scatter of the points and is therefore probably 
significant, even though different temperature pat¬ 
terns were present when different hydrophone depths 
were used. 

The corresponding plot for D> between 40 and 80 ft 
has already been given in Figure 30. For such tem¬ 
perature structure, the intensity first decreases with 
increasing depth as the hydrophone goes below the 


thermocline, and then increases. As pointed out in 
Section 5.3.4, the transmission anomaly below a sharp 
thermocline is likely to show better correlation with 
the depth and sharpness of the thermocline than with 
the temperature code used in Figures 49 and 51. In 
fact the limited results available are consistent with 
the belief that for MIKE and CHARLIE patterns in 
general the average transmission anomaly below a 
thermocline is approximately given by equation (13) 
in Section 5.3.4. 



Figure 52. Average transmission anomalies for 
hydrophone 50 to 100 feet deep. 


An example of the way in which transmission 
anomalies change with changing temperature struc¬ 
ture is shown by the set of average curves for hydro¬ 
phones between 50 and 100 ft deep shown in Figure 
52, again taken from reference 14. Most of the data 
used in these curves were actually obtained with 
hydrophones at 50 ft. The curves are in terms of the 
digits in the temperature-depth code explained in 
Section 5.1.4. The successive curves may be inter¬ 
preted as follows: 
d 2 d 3 

13 Temperature decreases to 0.3 F below surface tem¬ 
perature between 5 and 10 ft; between 20 and 40 ft it 
has decreased to 1 F below surface temperature. This is 
a moderately sharp NAN pattern with the sharp 
gradient extending practically up to the surface, and 


















































TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


133 



TEMPERATURE-F 
0 40 50 60 


I 200 


RANGE IN 


YAROS 


Figure 53. Sound channel ray diagram, extreme case. 


shows the corresponding rapid rise in the transmission 
anomaly. 

23 This is a much the same as above except that the gradient 
in the upper 10 ft is somewhat weaker. This curve and 
the preceding one are closely similar. 

33 This is a NAN, MIKE, or CHARLIE pattern, but in 
any case the thermoclinc is shallow and the attenua¬ 
tion high. 

34 The main thermocline is deeper here since the tem¬ 
perature is within 1 F of the surface temperature down 
to at least 40 ft. The hydrophone may be below or 
above the thermocline. 

35 The top of the main thermocline is not much above 
80 ft, and the hydrophone is either close to the top or 
above it. However, there are gentle gradients above 
the hydrophone, and these act to reduce the sound 
intensity. The reduction in sound intensity produced 
by the weak gradient between the projector and the 
hydrophone may be regarded as an example of layer 
effect. 

45 The gradients above the hydrophone are weaker and 
transmission is improved. 

55 The water is virtually isothermal down to 80 ft, and 
the results discussed in Section 5.2 are applicable. The 
deviation of this curve from a straight line is probably 
not significant. 

Sound Channels 

When the sound velocity at the projector is less 
than the velocities above and below, rays leaving the 
projector at sufficiently small angles will, in theory, 
curve back and forth within two fixed depths of equal 
sound velocity, giving rise to the curious ray diagram 
shown for an extreme case in Figure 53. This situa¬ 
tion is called a sound channel, and should in theory 
give rise to high sound intensity at long ranges. When 
sharp negative temperature gradients are present 
over sharp positive gradients, such sound channels 
should be persistent and very marked. However, very 
few measurements have been made with positive 
temperature gradients present in the water. 

In the absence of positive temperature gradients, 
the effect of pressure on sound velocity can produce 
a positive velocity gradient below the projector. How¬ 
ever, this gradient is very small, and a temperature 


decrease of only 0.3 F (at 60 F) between the projector 
and the isothermal layer will bend the sound rays 
down so sharply that an isothermal layer 100 ft thick 
is required to bend the rays back up again. More¬ 
over, as a result of this small gradient , rays bent down 
into the isothermal layer would return to the surface 
only at ranges of many thousands of yards. This 
bending is so gradual that the presence of thermal 
microstructure might be expected to mask com- 
pletefy any sound channel effects resulting from up¬ 
ward bending in nearly isothermal water. However, 
since some striking acoustic effects are observed with 
shallow gradients overlying isothermal layers, and 
since thermal microstructure has never been measured 
under such conditions, it is instructive to examine 
what the sound field would be like in truly isothermal 
water underlying slight gradients at the surface. 

If the projector were in such a hypothetical layer 
of completely isothermal water, the effects of the 
sound channel would not be particularly noticeable 
since the sound that, has curved first up into the 
negative gradient and then down into the isothermal 
layer would be indistinguishable from the rays that 
have traveled through the isothermal layer for their 
entire path. In fact downward bending by a very 
shallow surface gradient above the projector is proba¬ 
bly very similar to reflection by the surface. 

To produce marked effects the negative tempera¬ 
ture gradient at the surface must extend below the 
projector depth, so that the entire sound beam is bent 
downward, resulting in low sound intensities meas¬ 
ured by a shallow hydrophone at short range. Then 
when the rays are curved back to the surface thou¬ 
sands of yards out, the sound intensity should show 
a marked increase. On the basis of the simple ray 
theory, which neglects thermal microstructure and 
diffraction, the theoretical intensity at the projector 
depth is infinite at the range where the axial ray from 
the projector becomes horizontal again; this singu¬ 
larity results from the crossing of many adjacent rays 
at this point. Although of course diffraction and ther- 









134 


DEEP-WATER TRANSMISSION 




4890 4940 4990 

SOUND VELOCITY IN FT PER SECOND 


DATE 

TIME 

11-29-1943 

1400 

BT CLASS CHARLIE 
WATER DEPTH 8 80 FM 

SEA 

SWELL 

WIND 

1 

2 

FORCE 1 


Figure 54. Peaked transmission anomaly possibly resulting from sound channel ray diagram. 


mal irregularities will reduce this theoretical inten¬ 
sity, sound intensities considerably above normal 
would be expected at certain ranges if a sound chan¬ 
nel were produced by a slight temperature gradient 
lying above rigorously isothermal water. 

The conditions for a sound channel, when no posi¬ 
tive temperature or salinity gradients are present, are 
thus rather critical. There must be a temperature 
gradient at the surface which extends somewhat be¬ 
low the projector depth. Below this must be a layer 
of completely isothermal water a hundred feet or 
more in depth. The temperature difference between 
the isothermal layer and projector depth must be not 
less than about 0.1 F but not greater than about 
0.3 F, the exact limits depending on the surface tem¬ 
perature as well as on the depth of the layer. Thus, 
even if no thermal microstructure were ever present, 
sound channels of this type would normally be quite 
transitory, appearing during the development of sur¬ 
face heating in deep isothermal water; they would be 
expected to become prominent when the temperature 
difference between the projector and the isothermal 
layer increased to 0.1 F, and to disappear as the 
gradient extended downward and the temperature 


at projector depth gradually increased by another 
one or two tenths of a degree. 

Sound transmission measurements at the UCDWR 
laboratory in San Diego show transitory effects simi¬ 
lar to those which may be expected to result from 
sound channels. Because of the difficulty in reading 
the bathythermograph slide accurately to 0.1 F, and 
because of the high variability of thermal conditions 
in space as well as in time, it is not possible to predict 
from the bathythermograms exactly when a sound 
channel may be present. However, marked peaks in 
the measured transmission anomalies are occasionally 
found when thermal conditions are appropriate. 

A good example of the type of effect that can be 
observed is shown in Figure 54. As shown in the ac¬ 
companying temperature-depth record, a sharp nega¬ 
tive gradient extends from about 15 ft to the surface 
while below this a nearly isothermal layer extends 
down to 100 ft. A careful reading of the trace indi¬ 
cated a slight negative gradient in this layer (about 
0.3 F in 100 ft) giving a constant sound velocity be¬ 
low the projector. Moreover, the sharp surface gradi¬ 
ent did not extend below the projector depth. Thus, 
the ray diagram in Figure 54 does not show a sound 







































TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 


135 



BT INFORMATION 



/ 

; 

» 

V 

i 

i 

7 

-SENDI 

-RECEI 

1 

NG SHIP | 
VING SHIP| 

J 

1 

_I_ 



/ 

/ 

$ 

\ 

/ 

/ 

/ 


SOUND FIELD DATA 4840 4890 4940 

SOUND VELOCITY IN FT PER SECOND 



DATE 

3-22-1944 

TIME 

1600 

BT CLASS NAN 

WATER 

DEPTH 650 FM 

SEA 

1 

SWELL 

1 

Wl ND 

FORCE 1 




RANGE IN YARDS 


Figure 55. Peaked transmission anomaly with sound channel unlikely. 


channel although barely perceptible changes in the 
temperature-depth record would make it so. Never¬ 
theless, the anomalously high intensity at long range 
in the shallow hydrophone is very striking. If an 
absorption coefficient of 4 db per kyd is assumed, the 
increase of intensity at 6,000 yd shown in Figure 54 
is about 25 db. Until a more accurate means for de¬ 
termining ocean temperatures is available, it cannot 
be decided whether peaks of this type are actually 
the result of the focusing action predicted in sound 
channel theory. 

However, it is suggestive that an examination of 
all cases in which the anomaly curves show peaks of 
10 db or more shows that the temperature-depth 
curves for most of these are very similar to the curves 
which would, on the simple theory, give rise to sound 
channels. Out of the many hundreds of runs made off 
San Diego, only about 25 show these peaks. In almost 
all of these the thermocline is below 100 ft with nearly 
isothermal water above, and the receiving hydro¬ 
phone is at shallow depth. Moreover, in most cases 
slight negative gradients are present close to the sur¬ 


face. A few significant exceptions are present, as for 
example the run shown in Figure 55 where the deep 
hydrophone shows a peak of 20 db at 4,500 yd. As an 
example of the transitory nature of most such peaks, 
Figure 55 may be compared with Figure 42, which 
plots the run immediately preceding and shows no 
trace of any peaks. Also on one day (March 15, 1944) 
the shallow hydrophone showed a marked peak 
throughout the day while the temperature records 
only occasionally showed the deep layer of constant 
temperature required for a sound channel. Thus, 
while the evidence suggests that sound channels may 
in fact occur, there may quite possibly be other 
factors, still unexplored, that play a part in producing 
anomalously high intensities at certain ranges. 

5.4.3 Transmission at 60 kc 

The effects produced by temperature gradients on 
the transmission of underwater sound have been 
thoroughly explored only at 24 kc. A few measure¬ 
ments are available, however, at 60 kc. 
















































136 


DEEP-WATER TRANSMISSION 



Figure 56. Average transmission anomalies at 60 kc. 


Average transmission anomalies for a shallow 
hydrophone at 60 kc are shown in Figure 56 for dif¬ 
ferent values of D 2 . All runs for D 2 between 20 and 
160 ft are combined in the upper curve, since no 
systematic variation of transmission anomaly with 
changing D-> was noted for these data. The agreement 
between the curves for D 2 between 20 and 40 ft and 
for greater D 2 is in marked contrast to the differences 
shown at 24 kc (see Figure 49). 

Another, more conclusive indication of the compli¬ 
cated differences between the two frequencies is shown 
by simultaneous measurements at both frequencies on 
a single shallow hydrophone. Measurements were 
made with a CHARLIE pattern (temperature dif¬ 
ference about 0.5 F in the top 30 ft) and a thermoeline 



Figure 57. Simultaneous transmission at 24 and 
60 kc. 


at 50 ft. The resulting curves are shown in Figure 57. 
The difference of anomalies between the two fre¬ 
quencies increases by 15 to 20 db per kyd, as com¬ 
pared to a corresponding difference of not more than 
about 10 db per kyd in isothermal water; see Figure 
17. If data from many more runs under somewhat 
similar conditions are used, however, the average dif¬ 
ference of transmission anomaly between 60 and 24 
kc has a slope of about 9 db per kyd. This is in close 
agreement with the difference of 8.5 db per kyd found 
in Section 5.2.2 for isothermal water. Further data 
on the difference of transmission loss between dif¬ 
ferent frequencies are required to show how much and 
in what way this difference varies. 




















Chapter 6 

SHALLOW-WATER TRANSMISSION 


I n chapter 5, it was shown how the propagation 
of sound in deep water is affected by temperature 
gradients in the sea and by the sound frequency. 
In shallow water, these factors continue to operate; 
added to them is the effect of the bottom. The bottom 
affects the sound field in two different ways. Some of 
the sound incident on the bottom will be reflected and 
may penetrate into shadow zones. Also, some of the 
sound incident on the bottom will be scattered back¬ 
ward and will form part of the reverberation back¬ 
ground against which an echo must be recognized in 
echo ranging. This latter effect of the bottom will be 
considered in Chapters 11 to 17 of this volume. In 
this chapter, only the transmitted sound reaching a 
receiving hydrophone will be considered. 

6.1 PRELIMINARY CONSIDERATIONS 

In deep water, it was found that the most impor¬ 
tant single factor determining the transmission of 
sound of a given frequency is the vertical temperature 
structure of the ocean. The roughness of the surface 
of the sea plays a poor second, and nothing is known 
concerning the effects of other oceanographic varia¬ 
bles on sound transmission. In shallow water, the 
number of factors which may conceivably affect 
sound transmission is greater; it would be impractical 
to make a large number of sound transmission runs 
and then obtain rules of sound propagation empiri¬ 
cally merely by subjecting the data amassed to an 
unprejudiced statistical analysis. Rather, it was found 
necessary to assess beforehand the possible effects of 
bottom character, roughness of the sea surface, and 
refraction conditions, and then to analyze the trans¬ 
mission run data purposefully. This procedure proved 
successful in bringing order into a mass of data, and 
it will also be followed in this discussion. 

6.1.1 Effects of Sea Bottom 

If bottom-reflected sound is added to the sound 
field which reaches the receiving hydrophone (or the 
target in echo ranging), interference between the 
signals transmitted via the different possible paths 


may be either constructive or destructive, depending 
on the geometry of the paths. However, if the sound 
field intensity is averaged over a volume of the ocean 
sufficiently large to include several maxima and 
minima of the interference pattern, the averaged 
sound field intensity will be the algebraic sum of the 
intensities of sound resulting from each path by itself. 
In this sense, averaged sound field intensities in 
shallow water are always higher than sound field 
intensities in deep water under otherwise identical 
conditions. The extent to which bottom-reflected 
sound will increase the “deep w r ater” sound field in¬ 
tensity and to which it will eliminate shadow zones 
depends on a number of factors, which will be treated 
in this chapter. One of these factors is the reflectivity 
of the sea bottom. 

A theoretical treatment of bounding surfaces indi¬ 
cates that the reflectivity of a surface is determined 
by two factors: the degree of roughness of the surface 
itself, and the density and elastic moduli of the two 
adjoining media, such as sea water and granite. For 
the special case of two fluid media, it was shown in 
Section 2.6.2 that the percentage y e of reflected 
energy depends on the ratio of the densities as well 
as the angles of incidence and refraction, according 
to the formula 

PiCi — pcV" 1 + tan 2 0 (1 — Ci/c 2 ) 2 

- y e — - >- = (1) 

_PiCi + pcv 1 + tan 2 0 (1 — c?/c 2 )_ 

in which p and p x are the densities of the two adjoining 
media, c and Ci are the sound velocities, and 6 is the 
angle of incidence. This quantity y e is called the coef¬ 
ficient of reflection of the separating surface. The 
coefficient of reflection equals unity when the angle 
of incidence exceeds the critical angle for total re¬ 
flection. 

Equation (1) is based on the assumption that a 
smooth plane interface separates two perfect fluids. 
This assumption is not entirely correct for either the 
surface or the bottom of the ocean. The surface of the 
ocean is not smooth. With high winds, it may contain 
a large number of air bubbles (whitecaps), which 
absorb and scatter sound. The bottom often consists 


137 







138 


SHALLOW-WATER TRANSMISSION 


of material capable of shear stress, like rock, and is 
frequently rough. It is, therefore, simpler to deter¬ 
mine the coefficient of reflection experimentally than 
to attempt to obtain it from the mechanical param¬ 
eters of the two substances separated by the inter¬ 
face. 

A rough bottom will not only give rise to a trans¬ 
mitted sound wave, which disappears from the ocean, 
and a specularly 11 reflected wave, but will also scatter 
sound in a random fashion. The sound beam resulting 
from the reflection of a plane wave incident on a 
bottom of moderate roughness has a certain direc¬ 
tivity pattern. If the roughness is not excessive, this 
directivity pattern will show an intensity maximum 
in the direction which corresponds to specular re¬ 
flection from the bottom. The smoother the bottom, 
the more highly directive or collimated is the re¬ 
flected sound beam. 

Any nonspecularity of the reflection at the bottom 
has essentially the same effect as would a broadening 
of the directivity pattern of the projected beam. This 
increased divergence will not, however, always cause 
a decrease in the peak level of the received signal. If 
the bottom is smooth or only moderately rough, and 
if projector and receiver are not too highly directional, 
there should be little or no decrease in the peak signal 
level. This is because, on the average, as much energy 
will be deflected toward the hydrophone by non- 
specular reflection as will be lost out of the main beam 
through the same mechanism. Excessive roughness 
of the bottom, however, should cause a decrease in 
the peak signal level unless the projector is nondirec- 
tional and a long pulse is used. The reason for speci¬ 
fying a long pulse is that some of the sound scattered 
toward the hydrophone by a very rough bottom will 
travel paths much longer than the path correspond¬ 
ing to specular reflection. Thus, when short pulses of 
sound are projected over a rough bottom, the re¬ 
ceived signal will last longer than the transmitted 
signal. Although the total energy received at the 
hydrophone may be the same as if the bottom were 
smooth, the peak signal level will be lower. 

6 . 1.2 Velocity Gradients and 
Wind Force 

The magnitude of the contribution of bottom-re¬ 
flected sound to the total sound field will depend not 
only on the acoustic properties of the bottom, but 

0 Specular reflection is reflection for which angles of inci¬ 
dence and reflection are equal. 


also on refraction conditions in the body of the sea 
and on the reflectivity of the sea surface. 

The bottom should probably be of little importance 
if upward refraction in the sea volume prevented 
most of the sound energy from ever reaching the 
bottom. We may therefore tentatively predict that 
in the presence of positive gradients there will be no 
difference between deep-water transmission and shal¬ 
low-water transmission. On the other hand, in the 
presence of downward refraction the bottom should 
usually play a role of some importance. If the bottom 
is a very poor reflector of sound, then the sound field 
should not differ significantly from the sound field in 
deep water under similar refraction conditions, since 
bottom-reflected sound will make only a slight con¬ 
tribution to the total sound field. But if the ocean 
bottom is a good reflector, then the contribution of 
bottom-reflected sound will be significant. This con¬ 
tribution will increase in importance as the down¬ 
ward refraction becomes sharper and removes more 
and more energy from the direct sound field at long 
range. 

In addition, if the ocean bottom reflects sound 
fairly well, the sound field intensity at long range will 
probably be increased appreciably by sound which 
has been reflected several times between the ocean 
surface and the ocean bottom. We should, therefore, 
look for a dependence of the shallow-water sound field 
on the roughness of the sea surface. 

The quantitative prediction of sound field levels in 
shallow water, by combining the information on bot¬ 
tom and surface reflectivity with that on refraction 
conditions, would be very difficult. The bulk of the 
reliable information on shallow-water transmission 
has been obtained directly by means of transmission 
runs. The qualitative considerations of this section, 
however, have been valuable in planning these trans¬ 
mission runs and in interpreting the resulting sound 
field data. 

6.1.3 Effects of Frequency on 
Spreading Factor 

It was shown in Section 5.2.2 that the attenuation 
of sound in deep water depends strongly on the 
frequency. It has been tentatively suggested that the 
observed dependence of attenuation on frequency 
might be fitted by a 1.4th power law. 1 It might be 
expected that a formula of the form 

H = aR + 20 log R (2) 

would not be applicable for transmission in shallow 




SUPERSONIC TRANSMISSION 


139 


water since this formula was derived in Section 5.2.2 
without taking into account the contribution of sound 
reflected from bottom and surface. For the higher 
supersonic frequencies, this fear is frequently un¬ 
justified. As a matter of fact, in the presence of a 
well-reflecting bottom, equation (2) provides a better 
fit to the observations, considering all types of re¬ 
fraction conditions, than in deep water. This appar¬ 
ent paradox can be explained easily. For 24-kc sound, 
for instance, it is known that the attenuation in the 
direct beam amounts to 4 or 5 db per kyd. Beyond a 
range of 2,000 yd, the second term in the expression 
(2) increases less rapidly than the first term. As a 
result, modifications in the second term of equation 
(2) due to changes in the geometry of spreading are 
insignificant compared with the first, or absorption, 
term — at least at ranges where the effect of the 
bottom might be expected to become noticeable. On 
the other hand, deviations from equation (2) in deep 
water are common in the presence of downward re¬ 
fraction in the shadow zone. These deviations are 
mitigated by the appearance of bottom-reflected 
sound in the shallow-water sound field at long range. 
It is, therefore, convenient to plot and to analyze 
transmission anomaly in supersonic shallow-water 
transmission, since at short range the sharp inverse 
square drop is taken out of the transmission loss, 
while at long range the variations of transmission loss 
and transmission anomaly are not very different (see 
Figure 1 of Chapter 4). 

At sonic frequencies the situation is different, since 
the attenuation as determined in deep-water trans¬ 
mission experiments is very small, certainly less than 
1 db per kyd. As a result, the first term of expres¬ 
sion (2) does not overpower the second term even at 
ranges of the order of 10,000 yd. Moreover, sonic 
sources and receivers tend to be nondirectional, and 
bottom-reflected sonic sound tends to become im¬ 
portant at shorter ranges than does bottom-reflected 
supersonic sound. It may, therefore, be expected that 
the contribution of bottom-reflected sound will sig¬ 
nificantly affect sonic transmission at all ranges of 
operational importance for all refraction conditions 
except sharp upward refraction. At sonic frequencies, 
a modification of the inverse square law to take bot¬ 
tom-reflected sound into account thus is more neces¬ 
sary than at frequencies above 10 kc. If both the sur¬ 
face and the bottom were perfectly reflecting, sound 
energy would spread only in two dimensions, and as 
a result, the sound field decay at long range should be 
approximated by an inverse first power law. Actual 


interfaces permit sound to “leak” across, and the 
power law of sound field decay must be obtained by 
fitting a curve to the observations. Even under these 
circumstances, however, the consideration of trans¬ 
mission anomalies based on the inverse square law 
should reveal the essential features of sound trans¬ 
mission and may be preferred on the grounds of uni¬ 
formity of approach. In addition, a plot of transmis¬ 
sion anomaly has the practical advantage that a 
more open decibel scale is possible than for trans¬ 
mission loss. 

6.2 SUPERSONIC TRANSMISSION 

To study the acoustic properties of various sea 
bottoms, both UCDWR and WHOI have carried out 
transmission and reverberation runs in shallow water. 
The purpose of these experiments has been both to 
measure specific parameters characterizing the sea 
bottom and to obtain information on the overall 
properties of the sound field encountered in shallow 
water. Reverberation experiments are discussed in 
detail in Chapters 11 to 17 of this volume; but it is 
necessary to refer to them in this chapter, because 
they have incidentally furnished tentative values for 
the reflection coefficients of sea bottoms for slant 
incidence. 2 

6 . 2.1 Acoustic Properties of Sea 
Bottoms 

Types of Sea Bottoms 

Analysis of observed echo and listening ranges, 
which began in 1941, indicated that ocean bottoms 
could be roughly subdivided into a few geological 
types with fairly consistent reflection characteristics 
for each type. The classification of bottoms for sound 
ranging purposes has been standardized and includes 
the following: SAND, SAND-AND-MUD, MUD, 
ROCK, STONY, and CORAL. These bottom types 
are described as follows. 3 

SAND Firm, relatively smooth bottom. 

SAND-AND-MUD Relatively firm, smooth bottom. 

MUD Soft, smooth bottom. 

ROCK Rough, broken bottom. Includes 

bedrock, outcrops, and areas covered 
by boulders. 

STONY Hard bottom, commonly rough. Pre¬ 

dominantly cobbles, gravel, and 
shells. Varying amounts of sand and 
mud commonly present. 

CORAL Hard bottom, with sandy patches, 

irregular to smooth. Includes var¬ 
ious marine forms which secrete 
masses of lime covering the bottom. 



140 


SHALLOW-WATER TRANSMISSION 


In the actual classification of sea bottoms, the 
criterion established for estimating the relative firm¬ 
ness or softness of the bottom was grain size, as de¬ 
termined by mechanical analysis. The size limits 
were set as follows (Division 6, Volume 6). 

MUD 90 % by weight smaller than 0.062 mm. 

SAND-AND-MUD Between 10% and 90% smaller than 
0.062 mm. 

SAND Less than 10% smaller than 

0.062 mm and 90% smaller than 
2.0 mm. 

STONY Rounded or angular pieces of rock 

more than 2.0 mm and less than 
10 cm, which appear to represent 
glacial drift or other transported 
material. 

ROCK Rocks of a size greater than 10 cm 

or pieces broken from rock ledges or 
where bottom photographs show 
projecting rocks or rock ledges. 

CORAL Samples containing calcareous masses 

of coral, algae, or other lime secreting 
organisms, or bottom photographs 
showing their existence. 

This classification has been reasonably satisfactory 
from the acoustical standpoint except in the case of 
mud, for which it is now likely that texture alone is 
not an adequate criterion. 

Although the bulk of experimental work on sound 
transmission in shallow water has consisted of trans¬ 
mission runs, some special experiments have been 
made to determine numerical reflection coefficients 
of sea bottoms. 

Reflection Coefficients 

It may be gathered from the discussion in Section 
6.1.1 that different values of the reflection coefficient 
are to be expected for sound incident vertically on 
the bottom and for slant rays. Although the deter¬ 
mination of reflection coefficients for vertical inci¬ 
dence has some interest at sonic frequencies, the most 
important situations at all frequencies, from an opera¬ 
tional point of view, involve slant rays. To obtain the 
reflectivity of the sea bottom for slant incidence, 
three different experimental methods have been con¬ 
sidered. 

The most direct method uses transmission runs 
with very short signals (10 msec), which often permit 
the separate reception of the direct signal and the 
bottom-reflected signal. (The surface-reflected signal 
cannot be resolved for the usual projector depth of 
16 ft, but would be resolvable if the projector could 
be lowered to several hundred feet.) No coefficients 


of reflection resulting from these experiments have 
been reported. It is possible to make a very crude 
estimate of the reflection coefficient from published 
standard transmission runs in those cases where the 
curve showing the transmission anomaly plotted 
against range indicates at least two well-marked 
peaks corresponding to single and double bottom re¬ 
flections. Reading the level difference between con¬ 
secutive peaks and correcting for transmission loss 
due to absorption between reflection (say 4 or 5 db 
per kyd) leads to the following estimates: (1) for 
SAND, the loss through reflection amounts to be¬ 
tween 0 and 6 db per reflection, corresponding to in¬ 
tensity reflection coefficients between 1 and 0.25; (2) 
for MUD, the loss is between 10 and 30 db per reflec¬ 
tion, corresponding to coefficients of intensity reflec¬ 
tion between 10~ 3 and 10 -1 . The wide spread in each 
of these estimates indicates both the uncertainty of 
the estimate in a given case and the wide varia¬ 
bility among ocean bottoms falling within one clas¬ 
sification. 

The second method involves the measurement of 
bottom reverberation. If an echo-ranging transducer 
is tilted downward about 30 degrees, a peak of the 
reverberation is received at the range at. which the 
sound beam strikes the bottom. Sometimes, over 
well-reflecting bottoms, a secondary peak is observed, 
at a range at which the sound beam, specularly re¬ 
flected first by the bottom and then by the surface, 
strikes the bottom a second time. This secondary 
reverberation peak has been observed over a coarse 
sand bottom, and the average amplitudes of princi¬ 
pal and secondary peaks have been determined by 
UCDWR. 4 If reasonable assumptions are made con¬ 
cerning the transmission loss between the primary 
and the secondary reverberation peak, and if the re¬ 
flection at the sea surface is assumed to be perfect (a 
calm sea and a low wind), then the reflection coeffi¬ 
cient of the sea bottom can be estimated. It was found 
lie somewhere between 0.25 and 1.0, with the most 
probable value 0.5. These values were obtained for 
coarse sand, probably the bottom with the highest 
reflectivity. 

Reflection coefficients have been estimated as 0.031 
for foraminiferal SAND, between 0.005 and 0.025 for 
SAND-AND-MUD, and 0.0017 for MUD. 5 These 
determinations are not very reliable, because they 
involve unrealistic assumptions concerning the trans¬ 
mission loss between consecutive reflections. In these 
computations, it was assumed that the transmission 
anomaly amounted to 1.6 db per kyd of vertical path 



SUPERSONIC TRANSMISSION 


141 



SOUND FIELD DATA 


\ A 







A 

v 


V 

. /-\ 




• 

> 

DIRECT BEAM 

\ 

• 

\ 

\ 

FIRST BOTTO 

REFLECTION 

SECOND 
\ REFLEC 

hH 

30TT0M \ TH 
TION \ 

IRO BOTTOM 
REFLECTION 

\ 




\ 

• 



\ 


6 °0 2000 4000 6000 


RANGE IN YARDS 


SOUNO VELOCITY IN 
FEET PER SECOND 


DATE 8-6-1943 
TIME '330 _ 

BT Cl ASS NAN 
WATER TIFPTH 80 FM 

SEA_ I 

S W E LI_ 2 

WIND_ 2 


Figure 1 . Typical transmission run over SAXD bottom. 


traveled. This value is probably too low, and, there¬ 
fore, the reflection coefficients are too low, if the at¬ 
tenuation of vertical 24-kc pulses is anything like the 
attenuation of horizontal 24-kc pulses. 

WHOI has recently developed a new method for 
measuring bottom-reflection coefficients by means of 
a determination of the interference pattern found at 
short ranges when both transmitter and receiver are 
very close to the bottom. 6 This experiment is carried 
out with a cable-suspended transmitter which must 
be nondirectional in the horizontal plane, for the same 
reason that cable-suspended receiving hydrophones 
must be nondirectional in the horizontal plane (see 
Chapter 4). If the hydrophone is moved up and 
down while a constant distance is maintained be¬ 
tween transmitter and hydrophone, then the output 
of the hydrophone goes through a series of maxima 
and minima. If the bottom were specularly reflecting, 


then the ratio between the pressure at the maxima 
and minima should equal (1 + y a )/(l — 7a), where 
7 a is the amplitude-reflection coefficient of the bot¬ 
tom; the amplitude-reflection coefficient is the square 
root of the intensity-reflection coefficient y e , which is 
usually employed. The method yields values for the 
reflection coefficient which may be too low if the re¬ 
flection from the bottom is not completely specular. 
Experiments over a SAND-AND-MUD bottom 
have led to a value of the amplitude-reflection coef¬ 
ficient of 0.3 + 0.1, corresponding to an intensity- 
reflection coefficient of about 0.1. 

6 . 2.2 Analysis of 24-kc Trans¬ 
mission Runs 

Transmission runs in shallow water and at super¬ 
sonic frequencies have been made by UCDWR since 
1943, and by WHOI since 1944. Runs carried out in 











































142 


SHALLOTV-M ATER TR ANSMISSION 


RAY DIAGRAM AND BOTTOM PROFILE 




RANGE IN YARDS 



4890 4940 4990 

SOUND VELOCITY IN 
FEET PER SECOND 


DATE 

9-17- 

1943 

TIME- 

1330 


BT CLASS 

NAN 

WATER 

DEPTH 

27 FM 

SF A 

2 


SWELL 

2 


WIND 

3 



Figure 2. Transmission run over sand showing linear transmission anomaly. 


the presence of negative gradients have furnished tlie 
most valuable information on the reflectivity of sea 
bottoms, because of the large contribution of bottom- 
reflected sound under these conditions, even at 
relatively short ranges. Fortunately, the Pacific 
Ocean off southern California has sharp thermoclines 
most of the year, and the bulk of shallow-water trans¬ 
mission runs by UCDWR off San Diego were made 
in the presence of downward refraction. Figure 1 is 
a data sheet from a typical transmission run in 
shallow water over a SAND bottom. On the sheet, 
the sound data are plotted as transmission anomaly 
against range.' 5 The transmission anomaly vs range di¬ 
agram would be a horizontal straight line if the sound 
field intensity obeyed the inverse square law. If the 
transmission obeyed a law of the form of equation (2), 
the transmission anomaly would be represented by 


b It will be recalled that transmission anomaly is defined as 
the excess of the transmission loss in decibels over the value 
computed in accordance with the inverse square law of 
spreading. 


a slanting straight line, whose slope would be a, the 
coefficient of attenuation. It has already been men¬ 
tioned that in many cases the transmission anomaly 
can be approximated reasonably well by a straight 
line. Such a straight line, fitted by inspection, is 
drawn as the dashed line of Figure 2. The slope of this 
line is approximately 4.5 db per kyd. 

Experience has shown that reasonably linear trans¬ 
mission anomalies are typical of well- and fairly well- 
reflecting bottoms. The slope of the transmission 
anomaly curve depends markedly on the degree of 
reflectivity of the sea bottom, at least for supersonic 
sound, but much less on the exact shape of the tem¬ 
perature distribution, as long as the downward re¬ 
fraction is strong enough to force the direct sound 
field out of the depth of the receiving hydrophone. 

Effect of Velocity Gradients 

This section deals with an analysis of several 
hundred shallow-water transmission runs, which 
were obtained by the UCDWR and by the WHOI 
















































SUPERSONIC TRANSMISSION 


143 


laboratory groups. Plots of these runs were analyzed 
with respect to three factors: refraction pattern, 
depth of the receiving hydrophone and bottom 
character. Two other parameters which are also 
significant, the water depth and wind force, were not 
taken into account in order not to split the sample 
into too many small divisions. 

No analysis is at present available concerning the 
effect of water depth on transmission. It is known 
that in very shallow water (5 fathoms) transmission 
is poorer in the presence of downward refraction than 
it is over the same type of bottom in deeper water. 7 
For the purposes of the analysis reported below, runs 
in water of less than 10 fathoms and runs in water of 
more than 200 fathoms have been omitted. As for 
wind force, a separate analysis has been made at 
UCDWR, which will be reported at the end of this 
section. 

The following types of bottoms have been treated 
separately: SAND, SAND-AND-MUD (including 
SAND-MUD and MUD-SAND), MUD (including 
only the soft muds), CLAY (the plastic muds), 
ROCK (including ROCK and CORAL), and STONY 
(including gravel, cobbles, and similar notations on 
the original sheets). Sand-and-shells was treated as 
SAND. 

The depths of the receiving hydrophone were sub¬ 
divided into three classes: shallow (0 to 16 ft), inter¬ 
mediate (17 to 100 ft), and deep (more than 100 ft). 
These classes were chosen for convenience and uni¬ 
formity. A division of hydrophone depths into depths 
above and below the thermocline might have been 
preferable from a theoretical point of view; but on 
many bathythermograph traces the location of the 
thermocline is not uniquely determined. Therefore, a 
more mechanical division on the basis of hydrophone 
depth in feet was decided on. 

The bathythermograph patterns were divided into 
the usual classes, described in Section 5.1.4 as NAN, 
CHARLIE, MIKE, and PETER. All patterns were 
classified as in deep water, that is, the classification 
BAKER (used for most conditions in shallow water) 
was never used. The MIKE patterns were subdivided 
into two classes, DEEP MIKE, consisting of all pat¬ 
terns in which the water was isothermal to at least 
100 ft below the surface, and SHALLOW MIKE, in¬ 
cluding all other MIKE cases. 0 In the case of certain 


c This division of the MIKE patterns was made for this 
analysis only. The designations DEEP MIKE and SHALLOW 
MIKE have no official standing in Xavy doctrine. 


well-reflecting bottoms, NAN and CHARLIE were 
combined into one class. 

As a preliminary step, median and quartile R i0 
ranges' 1 w r ere determined for all combinations of the 
three parameters considered in this analysis; quartiles 
were omitted wherever the number of runs was 7 or 
less. Table 1 lists the results obtained for the 
UCDWR runs in shallow water. Table 2 lists the 
results obtained for the WHOI runs available at the 
time of the analysis. 

For each class of runs, two figures are supplied in 
the upper right-hand corner of the box for median 
values of R i0 in order to indicate the size and extent 
of the sample. The first of these two figures is the 
total number of runs making up the sample. The 
second number, which shall be called the “adjusted 
number of days” and is separated from the total 
number of runs by a slant line, indicates how widely 
distributed the sample is in time. The latter figure is 
supplied because it has been found that the acoustic 
data obtained on a particular day and at a particular 
location resemble each other more closely than data 
which have been obtained on different days, even 
though the oceanographic conditions are closely 
similar. Instead of simply noting the number of dif¬ 
ferent days on which the various runs making up the 
sample were obtained, it was decided to give an “ad¬ 
justed” number, computed as follows. If the number 
of runs made on k different days are denoted by n i, 
n 2 , - ■ ■ ,rik, then the adjusted number of days K is 
defined as the expression 


i = 1 

K equals the number of days k if all the rii are very 
nearly equal; in other words, if the sample is evenly 
distributed over the various days on which runs in 
this classification were obtained. But if some of the 
days furnish only one or two runs with other days 
contributing large numbers of runs, the days with 
very few runs will not be counted fully. To give a 


d These ranges represent that range at which the trans¬ 
mitted sound level is 40 db below the level at 100 yd. In some 
cases, the level at 100 yd was ascertained by extrapolating in 
from several hundred yards. In the ease of WHOI data, Rio is 
determined with reference to the sound level at 100 yd at the 
depth of the hydrophone in question, and R i0 is thus nothing 
but a measure of the slope of the transmission anomaly vs 
range; while for UCDWR data, reference is made to the sound 
level at 100 yd at a depth of 16 ft below the sea surface. 






144 


SHALLOW-WATER TRANSMISSION 




Upper 

quar- 

tile 

1,775 

1,700 

1,910 

2,500 


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3,130 

2,185 

3,810 


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Table 2. Median and quartile values of R 40 for WHOI transmission runs at 24 kc in shallow water. 


SUPERSONIC TRANSMISSION 


145 


MUD 

Upper 

quar¬ 

tile 

1,113 

1.838 




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CO 

Cl 

G 

to 

CM 

GO 

CO ^ 

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w* -«r 

tO o 

CM 

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r—> 


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Cl 

1^ 

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to 

r—t 

3,400' /l -° 

Lower 

quar¬ 

tile 

009 

837 




SAND- 

AND- 

MUD 

Upper 

quar¬ 

tile 

o 

tO 

tO 

1,450 




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CO 

- CO 

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—_ © 

Cl 

Cl 

d 

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•O 

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CO 00 

cm' r-T 

CM CM 

tO © 

L- tO 

L- L- 

of cm' 

Lower 

quar¬ 

tile 

tO 

CM 

© 

1,150 




STONY 

Upper 

quar¬ 

tile 






Median 



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tO 

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of 

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CM CM 

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co co' 

Lower 

quar¬ 

tile 






ROCK 

Upper 

quar¬ 

tile 






Median 



r~ ao 

Cl — 

-T CO 

tO o 

CM o 

ZD © 

CM —’' 



Lower 

quar¬ 

tile 






SAND 

Upper 

quar¬ 

tile 



2,500 

2,575 



Median 



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cm' 

Lower 

quar¬ 

tile 



1,275 

1,900 




Hydrophone 

depth 

Shallow 

Intermediate 

Deep 

Shallow 

Intermediate 

Deep 

Shallow 

Intermediate 

Deep 

Shallow 

Intermediate 

Deep 

Shallow 

Intermediate 

Deep 

BT 

Pattern 

NAN 

CHARLIE 

NAN 

AND 

CHARLIE 

SHALLOW 

MIKE 

DEEP 

MIKE 

















































































































146 


SHALLOW-WATER TRANSMISSION 


numerical example, assume that a total number of 26 
runs were obtained on 4 days. If the numbers of runs 
on these four days were 7, 7, 6, and 6, then the value 
of K for this case would be 4.0. If, on the other hand, 
two of the four days contributed 12 runs each, and 
the other two days only one run apiece, K would be 
found to equal 2.3. 

The results in Tables I and 2 indicate that SAND, 
ROCK, and STONY are well-reflecting bottom types; 
they lead to values of R i0 in excess of 2,000 yd in the 
great majority of cases, regardless of refraction con¬ 
ditions. CLAY also appears to be a well-reflecting 
bottom, although it should be emphasized that all the 
C'LAY runs by UCDWR were made at a single loca¬ 
tion off San Francisco, and that a generalization of 
the results obtained should be based on a more ade¬ 
quate sample. In this method of analysis, no sys¬ 
tematic dependence on hydrophone depth can be dis¬ 
covered, although the samples for deep hydrophones 
are mostly too small for the results to be considered 
conclusive. Transmission over STONY bottoms ap¬ 
pears somewhat better than over any other type of 
bottom. 

For the classification MUD, it is apparent that the 
dependence of R w on the conditions of refraction is 
similar to the situation in deep water. The WHOI 
SAND-AND-MUD data resemble the MLTD data 
in this respect . The R^ ranges are long when the water 
is isothermal, and short when the sound beam is bent 
downward by negative temperature gradients. In the 
classification SHALLOW MIKE, there is some evi¬ 
dence of layer effect over the poorly reflecting bot¬ 
toms. Layer effect is much weaker, if present at all, 
for SAND, ROCK, and STONY bottoms. There is 
also some evidence that in the case of MUD bottoms 
the transmission for the deep hydrophone is better 
than for the intermediate and shallow hydrophones. 
The transmission results obtained by UCDWR and 
by WHOI appear to be in fair agreement with each 
other except for the SAND-AND-MUD bottoms. 
In this classification, the transmission observed by 
WHOI is significantly poorer than the transmission 
observed by UCDWR. This disparity is not too sur¬ 
prising in view of the fact that the SAND-AND- 
MUD classification covers a wide variety of bottoms, 
namely, all those bottoms in which very fine particles 
are mixed with sand grains and in which the per¬ 
centage of sand grains lies between 10 and 90 per 
cent. It appears reasonable to assume that the SAND- 
AND-MUD bottoms investigated by UCDWR con¬ 
tained a larger percentage of sand grains and were 




\*HOI^ 




UC 




500 

1 




C MEOIAN CURVES 

ANO SUPERIMPO 

COINCIOE AT 15 

_ 

SHIFTEO 

SEO TO 

OO YARDS 

UCDWR 

WHOI 


Figure 3. Comparison of WHOI and UCDWR trans¬ 
mission data over sand with downward refraction. 


harder, on the average, than the SAND-AND-MUD 
bottoms investigated by WHOI. 

The value of R w is a useful parameter for the 
description of transmission; but in view of the fre¬ 
quent nonlinearity of the transmission anomaly 
range curve, no single parameter can be relied on to 
adequately characterize the transmission from very 
short to very long ranges. A more adequate method 
for describing a sample of transmission curves is the 




























SUPERSONIC TRANSMISSION 


147 


computation of a curve of median transmission 
anomaly. To obtain such a curve, values of the trans¬ 
mission anomaly are read off at a number of prede¬ 
termined ranges, such as every 500 yd. At each range, 
the median transmission anomaly is noted. If these 
median values are plotted against the range and the 
resulting points connected, the resulting curve will 
have the property that at each of the ranges at which 
values were read it separates the actual curves into 
two equally numerous portions. In a similar fashion, 
upper and lower quartile curves of transmission 
anomaly may be obtained. 

In general, the median anomaly curve will look 
smoother than the individual curves making up the 
sample, and “bumps” in the individual curves will 
not show up it they do not all appear at the same 
range. However, the median curve will be repre¬ 
sentative of average transmission conditions, and the 
quartile curves will provide a graphical measure of 
the spread of the sample. 

Obviously, median curves of transmission anomaly 
are valuable only if they are based on a fair-sized 
sample. For this reason, separate median curves were 
not constructed for all the classifications which were 
established in the analysis reported here. 

Not all individual transmission curves extend out 
to the same range. When transmission conditions are 
relatively poor, the reading of the traces must be 
stopped at a rather short range. When an appreciable 
number of curves in the sample cannot be read at 
the longer ranges, the median curve for the remainder 
apparently turns upward; since this upward turn has 
no physical significance, the median curve is stopped 
short in such cases. 

In the computations summarized in this chapter, 
median and quartile transmission anomalies were 
determined at 1,000 yd and every 500 yd from there 
on; the number of curves in the sample was noted at 
each of these ranges. 

As a first problem, the degree of consistency be¬ 
tween the UCDWR and the WIIOI runs was investi¬ 
gated. For this purpose, all runs over SAND with 
downward refraction (NAN and CHARLIE), re¬ 
gardless of hydrophone depth, were collected for each 
institution separately, and median and quartile 
curves of transmission anomaly plotted. These curves 
are shown in parts (.4) and ( B ) of Figure 3. There is 
some evidence that the discrepancy of about 5 db be¬ 
tween the curves is due, in part, to a different method 
of calibration. While UCDWR has usually referred 
transmission anomalies to the transmission level re- 




Figure 4. Comparison of UCDWR and WHOI trans¬ 
mission data over sand-and-mud bottoms with strong 
downward refraction. 

corded at about 100 yd, WHOI has frequently ob¬ 
tained the reference level by extrapolating the meas¬ 
ured relative transmission anomalies backward. To 
illustrate the relatively good fit which results from 
vertical shifting of the curve, part (C) of Figure 3 
shows the two median curves shifted so that they co¬ 
incide at a range of 1,500 yd. 


























148 


SHALLOW-WATER TRANSMISSION 



We have already noted, from consideration of the 
/?4o values, that for SAND-AND-MUD bottoms the 
agreement between WHOI and UCDWR is very 
poor. This discrepancy is confirmed by the median 
and quartile transmission curves of those runs over 
SANI)-AND MUD which were carried out with a 
shallow hydrophone in the presence of NAN pattern 
(strong downward refraction), shown in Figure 4. In 
this case, the discrepancy is undoubtedly real and 
not caused by different calibration methods; for not 
only are the transmission anomalies at a given range 
different, but the WHOI median curve has a much 
steeper slope. The slope of the median UCDWR is 
roughly between 8 and 10 db per 1,000 yd, while the 
slope of the WHOI median curve is about 18 db per 
1,000 yd. 

No other comparisons were made between WHOI 
and UCDWR transmission data because most of the 
WHOI samples were too small for such comparisons. 
All the median curves to be discussed later are based 
exclusively on UCDWR runs. 

To examine the effect of hydrophone depth over 
a well-reflecting bottom, median curves over SAND 
were determined for the three classes of hydrophone 
depth without regard to refraction pattern. In Fig¬ 
ure 5, the three resulting curves are superimposed on 


each other, identified as s (shallow), i (intermediate), 
and d (deep). Table 1 shows that the bulk of these 
runs were carried out in the presence of downward 
refraction, with about one-fourth of the BT patterns 
showing a shallow mixed layer above the thermocline. 
In Figure 5 there are no significant differences 
between the three curves. 

The quartile curves have not been reproduced, but 
they are all fairly similar, deviating from the median 
curve by about 5 db at 3,000 yd. The transmission 
anomaly over SAND can be represented fairly well 
by a straight line passing through zero at zero range 
and having a slope of 5 + 2 db per kyd. This 
numerical estimate is also good for the median and 
quartile curves shown in Figure 3, which do not in¬ 
clude the SHALLOW MIKE cases forming part of 
the sample used in constructing Figure 5. 

Figure 6 shows median curves, for shallow and for 
deep hydrophones, of all runs obtained over ROCK 
bottoms. It will be noted that, the transmission over 
ROCK is not quite so good as over SAND, the aver¬ 
age slope for ROCK being 6 db per kyd. The quartiles 
deviate from the median, in this case also, by roughly 
2 db per kyd. 

Figure 7 shows the median and quartile curves for 
all runs obtained by UCDWR over STONY bottoms. 











SUPERSONIC TRANSMISSION 


149 



Transmission appears to be better out to 2,000 yd for 
STONY bottoms than for any other type of bottom, 
but deteriorates rapidly from 2,000 yd on out. The 
wide spread between the median and quartile curves 
is an indication that these bottoms are acoustically 
less uniform than SAND or ROCK bottoms. 

Figures 8 and 9 show two typical transmission 
anomaly plots, which were obtained over a ROCK 
and over a STONY bottom respectively. These runs 
were carried out in the presence of pronounced 
negative gradients from the surface of the sea down 
to well below the depth of the projector. The ray 
diagrams, which are shown in the upper parts of the 
figures, indicate that in deep-water transmission con¬ 
ditions would be confidently predicted to be poor. 
Because of the well-reflecting bottom, however, the 
observed transmission is comparable to that in a 
deep mixed layer in deep water. 

MUD bottoms were originally defined as bottoms 
in which the average particle was too small in size to 
be classified as SAND. However, evidence accumu¬ 


lated indicating that from an acoustic point of view 
there are two different types of bottoms which are 
composed of very small particles. These two types of 
bottom can be characterized by their consistency as 
soft and as plastic, and they have been designated in 
this analysis as MUD and as CLAY. Some evidence 
concerning the difference in acoustic properties of 
these two types of bottoms has been collected and 
published by WHOI. 8 This evidence for separating 
MUD bottoms into MUD and CLAY was apparently 
borne out by the analysis of the transmission data 
obtained by UCDWR, but doubts as to the correct 
classification of the CLAY samples involved detract 
from the value of this evidence. 

Figure 10 shows median and quartile transmission 
curves over MUD in the presence of negative gradi¬ 
ents, separated according to hydrophone depth. It 
appears that the quartile spread is appreciably re¬ 
duced for the shallow and deep hydrophone depths 
by this separation. Regardless of hydrophone depth, 
the transmission anomaly at 3,000 yd is approxi- 












150 


SHALLOW-WATER TRANSMISSION 



mately 30 db. For shorter ranges, there is a significant 
difference between the anomalies at different hydro¬ 
phone depths. With the hydrophone deeper than 
100 ft, the transmission anomaly is almost linear and 
increases at the rate of about 10 db per kyd. For the 
more shallow hydrophone depths, there is a much 
more precipitate drop at short range. With the hydro¬ 
phone at 16 ft, the median transmission anomaly at 
1,000 yd is 26 db. From 1,000 to 3,000 yd, it drops 
only another 8 db, resembling in this respect the trans¬ 
mission of sound in the shadow zone in deep water. 

Figure 11 shows a typical run over a soft MUD 
bottom in the presence of a pronounced negative 
temperature gradient. Just as in deep water, the 
sound level at shallow depth begins to drop rapidly 
at a shorter range than does the level at considerable 
depth. At all hydrophone depths, the transmission 


anomaly increases sharply at the approximate range 
of the predicted shadow zone boundary. A slight 
recovery of the sound level recorded by the two 
shallow hydrophones is noted at almost exactly the 
range at which the axis of the reflected beam rises to 
the depth of the hydrophone. This recovery is, how¬ 
ever, not very pronounced. While it increases the 
sound level at 2,400 yd to approximately 10 db above 
the level which would have been recorded in deep 
water under similar circumstances, the transmission 
anomaly still amounts to about 25 db. 

Effect of Wind Force 

UCDWR has carried out an analysis of the effect 
which the roughness of the surface has on sound 
transmission in shallow water over well-reflecting 
bottoms such as ROCK and SAND. 















TRANSMISSION ANOMALY IN DB DEPTH IN FEET TRANSMISSION ANOMALY IN DB DEPTH IN FEET 


SUPERSONIC TRANSMISSION 


151 




O 2000 4000 6000 

RANGE IN YARDS 

Figure 8. Transmission run over ROCK. 


BT INFORMATION 



DATE 

5-27-1944 

TIME 

1430 

BT CLASS N AN 

WATER 

DEPTH 42 FM 

SFA _ 

1 

SWELL 

I 

WIND 

2 



RAY DIAGRAM AND BOTTOM PROFILE 


BT INFORMATION 



s---/- 

-X^ThVdr 
~X\r- HYD^ 

1 1 

OPHpNE DEP 
OpT?ONE DEP 

TH A- 

TH O- 

- -Z 

~/r 

RECEIVING 

END 


STONY (COBE 

ILES) 

SE 

INDING END 









SOUND FIELD DATA 



0 


UJ 

U) 


5 100 

I 

H 

Q. 

UJ 

O 


200 

4890 4940 4990 



1 

1 

-RECEIV 

-SENDIN 

ING VESSEL 

G VESSEL 


H 



SOUND VELOCITY IN 
FEET PER SECOND 


DATF 11-9-1943 
TIME 1440 

BT CLASS NAN 
WATER DEPTH 24 FM 

SEA_!_ 

SWELL I _ 

WIND_!_ 


Figure 9. Transmission run over a STONY bottom. 































































































152 


SHALLOW-WATER TRANSMISSION 



Table 3. R i[t 

versus wind force over 

ROCK. 


Wind force (Beaufort) 

1 

2 

3 

4 

Number of runs 

29 

54 

31 

21 

Lower quartile Rto 

2,050 

2,150 

1,550 

1,500 

Median f? 40 

2,950 

2,400 

2,100 

1,750 

Upper quartile R. m 

3,150 

2,650 

2,300 

2,100 


One hundred thirty-five runs were carried out over 
ROCK bottom at fairly constant depth. In Table 3 
are listed the number of runs made at each wind 
force and the median and quartile values of R i0 . 
Figure 12 shows the complete distribution. 

One hundred seventy-four runs were carried out 
over SAND bottoms in water of more than 6 fathoms. 
Table 4 shows the same data for these runs as 
Table 3 does for runs over ROCK. 



























SUPERSONIC TRANSMISSION 


153 



SOUND FIELD DATA 




50 60 TO 

TEMPERATURE F 


date 

6-1-1944 

TIME 

1415 

BT CLASS N AN 

WATER 

DEPTH 47 

«;fa 

1 

SWELL 

WIND 

2 

1 



Forty-six runs were made in water of 6 fathoms or 
less and over SAND bottoms. These runs were 
analyzed as a separate group. Table 5 summarizes 
the results. Figure 13 shows the complete distribu¬ 
tion of SAND runs. 

The majority of the runs used in this analysis were 
carried out in the presence of downward refraction, 
but no attempt was made to separate the runs with 
downward refraction from those with the projector 
located in a mixed layer. This method of analysis may 
account for the wide quartile spread and more par¬ 
ticularly for the great upper quartile spread for wind 
force 4. The reduction of R. w between wind force 0 or 1 
and 3 amounts to an increase in the slope of the trans¬ 
mission anomaly curve of roughly 1 db per kyd. 

6.2.3 Summary 

Transmission experiments at 24 kc indicate that 
the sea bottoms can be roughly divided into well- 
reflecting bottoms comprising ROCK, CORAL, 
STONY, SAND, and CLAY bottoms, and poorly 
reflecting bottoms, mostly MUD and some of the 
SAND-AND-MUD. Most of the SAND-AND- 
MUD bottoms are intermediate between well and 


Table 4. R i0 versus wind force over SAND in water 
depth greater than 6 fathoms. 


Wind force (Beaufort) 

0 1 

2 3 

4 

Number of runs 

18 33 

51 60 

12 

Lower quartile Rw 

2,850 3,100 

2,900 2,100 

1,400 

Median R.w 

3,450 3,250 

3,500 2,450 

1,800 

Upper quartile R in 

4,000 3,500 

3,700 2,950 

2,500 

Table 5. R \o versus wind force over SAND in water 
depth 6 fathoms or less. 

Wind force (Beaufort) 

2 

3 

4 

Number of runs 

16 

13 

17 

Lower quartile Rw 

1,400 

1,250 

950 

Median R ia 

1,550 

1,400 

1,100 

Upper quartile Rv, 

1,700 

1,750 

1,650 


poorly reflecting bottoms. Present evidence indicates 
that in shallow water at least 10 fathoms deep and in 
the presence of downward refraction, transmission 
anomalies over SAND and STONY bottoms increase 
with the range by 5 ± 2 db per kyd, and over 
ROCK bottoms by 6 + 2 db per kyd. The trans¬ 
mission is not significantly affected by hydrophone 





















































154 


SHALLOW-WATER TRANSMISSION 



Figure 12. Rm versus wind force over ROCK. 


depth or bottom depth. (In very shallow water, less 
than 10 fathoms deep, transmission is inferior to that 
found in moderately shallow water.) Transmission 
over MUD differs but little from transmission in 
deep water; secondary peaks due to bottom-re¬ 
flected sound are not likely to raise the level more 
than 10 db above the level that would be observed 
in a deep-water shadow zone. In isothermal water or 
with upward refraction, transmission over all bot¬ 
toms is about as good and sometimes slightly better 
than deep water. 

Transmission anomalies with negative gradients 
over the well-reflecting bottom types are affected ad¬ 
versely by heavy seas. For sea state 3, transmission 
anomalies are likely to be at least 1 db per kyd higher 
than in calmer seas. 

6.3 SONIC TRANSMISSION 

Sonic transmission differs from supersonic trans¬ 
mission primarily in that dissipative processes within 
the water are much less important. The probable 
value of the absorption or attenuation coefficient at 



Figure 13. R t0 versus wind force over SAND. 


sonic frequencies has been discussed in Chapter 5. 
It has been estimated 9 that at the lower sonic 
frequencies (2,000 c and less) the attenuation of 
sound in sea water at a depth of several hundred 
fathoms is less than 1 db in 20,000 yd. While there is 
reason to believe that close to the surface, absorption 
at these low frequencies is appreciably higher, e it is 
probably no more than about 0.5 db per 1,000 yd. As 
a result, sonic sound in shallow water may show 
evidence of a spread less than that predicted by the 
inverse square law. This section summarizes the re¬ 
sults which were obtained by UCDWR, 10-12 and by 
CUDWR-NLL. 1314 In these experiments, CUDWR- 
NLL used a single-frequency source, with higher 

e If dissipative processes near the surface at low sonic fre¬ 
quencies were as low as they were estimated at great depths 
in reference 10, then listening ranges on noisy surface targets 
should be of the order of 100 miles in the presence of deep 
mixed layers; actual listening ranges rarely exceed 20 miles 
even with the best sonic listening gear available. 



















































SONIC TRANSMISSION 


155 


harmonics present because of overloading, while 
UCDWR used a noise source of the type employed 
for acoustic minesweeping. The receiving equipment 
consisted of hydrophones, whose output was ampli¬ 
fied and sometimes put through band filters to be 
recorded by means of power level recorders. Trans¬ 
mission was always continuous throughout the run. 

6.3.1 Long Island Area Survey 

Long Island Sound is mostly shallow, less than 
15 fathoms deep, and the bottom is predominantly 
sandy, although some runs were made over MUD, 
SAND-AND-MUD, and STONY bottoms. All runs 
were made with a single-frequency source. Fre¬ 
quencies used were 0.6, 2, 8 , and 20 kc. Geographi¬ 
cally, the survey was divided into three areas: the 
Fisher’s Island area, the New York Harbor ap¬ 
proaches, and Block Island Sound. In all three 
areas, hard bottoms were predominant. Depths 
varied from about 50 ft to 200 ft. During the New 
York Harbor and Fisher’s Island area surveys, re¬ 
fraction was mostly upward, owing in part to salinity 
gradients. Off Block Island, some negative gradients 
were found. Sea states were low with the exception 
of the New York Harbor runs where sea states up to 
4 were encountered. Table 6 summarizes the results 
obtained. To obtain this table, the investigators at¬ 
tempted to fit each run by a formula of the form 

H = n -10 log r + ar (4) 

in which n is the power of spreading and n represents 
the attenuation in decibels per kiloyard. Since it is 
difficult to determine both n and a simultaneously by 
a best fit calculation, n was chosen arbitrarily to as¬ 
sume the values 1, 1.5, and 2. The best value of a was 
then determined by inspection for each of the three 
assumed values of n. 

The three fits for n = 1, 1.5, and 2 were classified 
in order of decreasing preference as I, II, III. In addi¬ 
tion, the individual fits were graded on an absolute 
standard as “good” (< 7 ), “fair” (/),“poor” (p). Table 7 
is a summary of the goodness of the fit obtained for 
these runs. Despite the equal standard deviation 
values of Table 6 , the value 1 for n seems to be most 
frequently the best fit to the data, although 1.5 is 
probably the best average, especially at the higher 
frequencies. 

This survey resulted in the following general con¬ 
clusions. Higher frequencies were attenuated more 
than the lower frequencies. At high frequencies the 
transmission loss increased with increased disturbance 


£ O 


L- Ol q CO 

^ oi co o 


q q 
oi d 


00 coqq (N H M 
C 4 h'n 10 O £ <N d 


q ooc -h ci H qq 
oi o co d o d 


<M Tf U- X <M 


£ 2 b 
c2 *> 

So 


c 


OOOO *0 Ol iC O N h 

oi co d^’oico 


cq o h oq q q q 
oi co o rH cc n 


N N NN q q iq CNMq 

d 0 ^ rji oi co d i-H 0$ Tf 


CO x o c: o: 10 N 


qqqq q q cq qoqoo 

r-J ^ r-H ^ H d rf ^ oi c4 CO 


x q q co qqq qqq oo 
h h n 10 co d d oi co d x 


NNOON q cq q qqqq 
co’t n ri oi dec 


£ ° 
£ o 


U- X X X 


O Ol X O OXO' OOIXO 


, HH 1—1 1—1 1—1 d d d d 

, Ksapq £ 


* Values of a and a are given in decibels per kiloyard. 



































156 


SHALLOW-WATER TRANSMISSION 


of the sea for upward refraction, but there was no such 
effect at the lower frequencies. With downward re¬ 
fraction the state of the sea did not influence the 
transmission. Except at the lowest frequencies and 
over the softest bottoms, the type of bottom did not 
appreciably affect the transmission loss. Bottom types 
were MUD, SAND, and GRAVEL. Shoal areas and 
areas over sea valleys showed high transmission 
losses. The attenuation was virtually independent of 
depth for flat bottoms. Some correlation was found 
between the empirical value of n and refraction con¬ 
ditions; the power of spreading tended to assume 
large values in isothermal water. 


Table 7. Summary of shallow-water results (BI,FI) 
number of fits in indicated classification. 


n 

Classified 

0.6 kc 

2.0 kc 

8.0 kc 

20 kc 

1.0 

I 

18 

7 

6 

18 


II 

5 

3 

9 

4 


III 

8 

1 

16 

13 


0 

12 

2 

11 

20 


f 

17 

9 

9 

10 


p 

2 

0 

11 

5 

1.5 

1 

9 

4 

15 

9 


II 

22 

7 

16 

26 


III 

0 

0 

0 

0 


y 

10 

1 

15 

18 


f 

16 

7 

11 

14 


p 

5 

3 

5 

3 

2.0 

I 

4 

0 

10 

8 


II 

3 

1 

6 

5 


in 

24 

10 

15 

22 


Q 

4 

0 

7 

6 


f 

11 

2 

18 

22 


P 

16 

9 

6 

7 


6 . 3.2 Pacific Ocean Measurements 

Twenty transmission runs were made in the coastal 
waters off volcanic islands in the Pacific (see refer¬ 
ence 11). Measurements were made of overall trans¬ 
mission in the 1- to 3-kc band over SAND and 
CORAL bottoms. These measurements permitted the 
following conclusions. In the 1- to 3-kc band, the 
transmission loss from 100 to 3,000 yd could best be 
fitted by n equal to 1.5 and a equal to 2.5 db per kyd, 
under most hydrographic conditions. These condi¬ 
tions included slight upward refraction in water of 
depth about 200 ft, slight upward refraction over 
sloping bottoms, and downward refraction in water 
of depth about 100 ft. For downward refraction over 
a sloping bottom, however, the transmission loss at 
ranges above 1,000 yd was much greater. For this 
case, the best fit above 1,000 yd was estimated to be 


10 db per kyd for the attenuation with the spreading 
factor n uncertain. This result is in agreement with 
ray theory, 15 which predicts that sound multiply re¬ 
flected from the bottom under these conditions should 
run downhill, following the bottom slope and leaving 
a shadow zone near the surface. 

Also, some runs were made in the Thirteenth Naval 
District. 11 These runs were made in water less than 
300 ft deep over coarse gravel or rocky bottoms. 
Velocity gradients were slight and the sea calm. No 
correlation with computed limiting ranges was ob¬ 
served. The majority of the runs were best approxi¬ 
mated by zero attenuation and n equal to the values 
given in Table 8. One run through a tide rip was best 
approximated with n equal to 1 and the values of a 
given on the right-hand side of Table 8. 


Table 8. Summary of shallow-water results (Thirteenth 
Naval District). 


Ordinary Runs 


Special run through 
tide rip 

Frequency 

Average n 

a 

n 

a 

0.1 

1.4 

0 

i 

3.0 

0.6 

1.3 

0 

i 

1.5 

2.0 

1.5 

0 

i 

3.5 

8.0 

2.5 

0 

i 

8.0 

20.0 

3.2 

0 

i 

8.0 


6.3.3 Summary 

Recent experiments carried out by UCDWR with 
pulses of sonic single-frequency sound have not yet 
been reported; they are, therefore, not included in 
this summary. This summary lists conclusions which 
were reached in the spring of 1945 on the basis of 
data available then. 1617 It should be pointed out, 
however, that none of the conclusions reached at that 
time have been invalidated by later information. 

In shallow water, a distinction must be made be¬ 
tween transmission over MUD bottoms (which re¬ 
sembles deep-water transmission) and transmission 
over all other bottom types. No significant differ¬ 
ences were discovered in sonic experiments between 
any of the other bottom types including MUD- 
AND-SAND. Over sloping bottoms, a significant 
dependence on refraction pattern has been observed: 
with downward refraction transmission tends to be 
poor, while in isothermal water it is as good as in deep 
water. 

Over level bottoms, with isothermal water or in the 
presence of downward refraction, the transmission 






















SONIC TRANSMISSION 


157 


loss can be most adequately represented by an equa¬ 
tion having the form, 

H = 15 log r + a, (5) 

where a, the coefficient of attenuation in decibels per 
kiloyarcl, depends on /, the frequency in kilocycles, 
according to 

a = 0.25(/ — 2) (6) 

above 2 kc. Below 2 kc the attenuation is very small. 
Equation (6) is believed to be adequate up to about 
20 kc. 

Equation (6) represents merely the average de¬ 
pendence of the attenuation coefficient on frequency. 


Table 9. Variation of attenuation with frequency. 


Source 

a at 
2.0 kc 

a at 
8.0 kc 

a at 
20.0 kc 

a at 

8.0 less 
a at 2.0 

a at 

20.0 less 
a at 2.0 

San Diego 

0.0 

2.5 

4.2 

2.5 

4.2 

Fisher’s Island 

0.6 

1.7 

4.1 

1.1 

3.5 

Block Island 

2.3 

3.6 

7.1 

1.3 

4.8 

New York Har¬ 
bor 

1.4 

3.1 

7.9 

1.7 

6.5 

Average 

1.1 

2.7 

5.8 

1.6 

4.7 


In the portion of the sea fairly near to the surface, 
which is the only region of interest in sonic listening, 
the absorption coefficient probably depends on highly 


variable factors, such as bubble content ; thus large 
deviations from equation (6) may be expected to 
occur quite frequently. 

There appears to be little correlation at sonic fre¬ 
quencies between transmission loss and refraction 
conditions, depth of the water, and surface rough¬ 
ness. With strong upward refraction, an increase of 
attenuation with increasing sea state has been ob¬ 
served, undoubtedly caused by the poor reflectivity 
of a rough and aerated surface. 

At short ranges, out to approximately the range 
equal to the depth of the water, image interference 
maxima and minima have frequently been measured. 
However, except possibly at very low frequencies, 
the inverse fourth power decay has not been observed 
because of the disruptive effect of bottom-reflected 
sound at the ranges where the fourth power decay 
might be expected. 

In general, reliable information on sonic transmis¬ 
sion is scanty and is less consistent than the informa¬ 
tion on the transmission of 24-kc sound. In the future 
an increasing amount of stress is likely to be laid on 
the investigation of sonic transmission. However, 
sonic transmission will probably remain a more 
difficult field for investigation than supersonic trans¬ 
mission, because of the low directivity of most 
sources of sonic sound. 

















Chapter 7 

INTENSITY FLUCTUATIONS 


I t is clear from the preceding chapters that the 
sonar officer or the research worker cannot pre¬ 
dict with precision the sound field intensity in the 
vicinity of a calibrated sound source, no matter how 
complete his information on oceanographic condi¬ 
tions. Chapter 5, in particular, mentions the wide 
range of sound field levels which are recorded under 
identical or nearly identical oceanographic condi¬ 
tions. 

This chapter will be concerned with the variability 
of the sound field which is found when a succession of 
single-frequency signals are transmitted over the 
same path and received and recorded through the 
same receiving sound head and stack. This variability 
within a single sequence of sound signals has been 
subdivided into fluctuation, changes in intensity ob¬ 
served to occur during seconds or fractions of a 
second; and variation, a slow drift of the average in¬ 
tensity, which becomes noticeable in the course of 
minutes. This division between short-term and long¬ 
term variability can be justified on practical grounds. 
Variation may well be correlated with those large- 
scale changes in the thermal structure of the ocean 
which would be revealed by a continuously recording 
bathythermograph. Fluctuation is caused by mecha¬ 
nisms which cannot be observed by means of any 
oceanographic instrument in current use. This chap¬ 
ter will be concerned, exclusively, with the short-term 
variability of the sound field. The longer-term varia¬ 
bility has already been discussed in Chapters 5 and 6. 

The first section of this chapter will set forth the 
mathematical concepts commonly used in the de¬ 
scription of fluctuation and will report the results of 
fluctuation experiments. In the second section, the 
significance of these experimental results will be as¬ 
sessed, and the contribution of various mechanisms 
to the observed fluctuation will be estimated tenta¬ 
tively. 

7.1 OBSERVED FLUCTUATION 

7 . 1.1 Magnitude of Fluctuation 

In describing fluctuation quantitatively, we need 
expressions which characterize both the magnitude 


of fluctuation — roughly the amount by which an 
individual signal deviates from the mean for the run 
— and the time rate at which the sound field in¬ 
tensity changes. This subsection will be concerned 
with the magnitude of fluctuation. 

Three different quantities are commonly used to 
express the magnitude of a received signal: the pres¬ 
sure amplitude (in dynes per square centimeter), the 
intensity (in watts per square centimeter), and the 
level (in decibels above some standard). When we 
consider a sequence of N signals received under ap¬ 
parently identical conditions, we can characterize 
this sequence by three sets of figures: amplitudes, in¬ 
tensities, and levels of all the individual members of 
the sample. Each of these three sets of figures de¬ 
scribes the sample. Depending on our particular 
viewpoint, we may prefer one or another. 

These three sets of figures can be converted one 
into another by means of the two equations 


L = 10 log — = 20 log — , (2) 

1 0 Oo 

in which a stands for the pressure amplitude, / for 
the intensity, and L for the level in decibels. To each 
of these three sets we may assign as an average 
quantity the arithmetical mean, such as 

a = — (oi + a 2 + • • • + a,v)) (3) 

and refer to these quantities as the mean amplitude, 
the mean intensity, and the mean level of the sample. 
These average quantities are no longer related by the 
equations (1) and (2). 

Individual amplitudes will, of course, deviate from 
the mean amplitude. But some of these deviations 
will be positive, others negative, and it can be shown 
very easily that their sum vanishes. To express the 
spread of the amplitudes of the sample about the 
mean amplitude, a common procedure is to square 
the deviation of each individual amplitude from the 
mean amplitude and to average these squared devia¬ 
tions. The square root of the mean of the squared 


158 


OBSERVED FLUCTUATION 


159 


deviations has the same dimension as an amplitude. 
It is called the root-mean-square (rms) deviation of 
the amplitude or, more briefly, the standard devia¬ 
tion of the amplitude. If it is divided by the mean 
amplitude, the resulting dimensionless quantity is 
called the relative standard deviation of the ampli¬ 
tude; this quantity is often expressed in per cent. 

The analogous quantities formed with intensities 
and levels bear analogous names. These names are, 
in fact, common in all fields of statistics. If the rela¬ 
tive standard deviation of the amplitude is very 
small compared with unity, the relative standard 
deviation of the intensity is about twice the relative 
standard deviation of the amplitude, while the abso¬ 
lute standard deviation of the level is approximately 
4.34 times the relative standard deviation of the in¬ 
tensity.' 1 The fluctuation of underwater sound is 
usually so large that these relationships between the 
standard deviations do not hold. 

Relative standard deviations of the amplitude 
have been determined for transmitted signals of 
underwater sound under various conditions. 1-3 Most 
of the available data were taken at 24 kc. From the 
data at that frequency, an analysis was made of the 
dependence of the relative standard deviation on 
refraction conditions. 2 It was found that in the 
presence of strong downward refraction the median 
of the relative standard deviation, for the 29 samples 
collected, was 38 per cent. 1. For eleven samples, in 
which the receiving hydrophone as well as the pro¬ 
jector were located within a mixed layer above a 
thermocline, the median of the relative standard 
deviation was 47 per cent. Seventeen samples, in 
which the hydrophone was in the thermocline be¬ 
neath a mixed layer, showed a median relative stand¬ 
ard deviation of 41 per cent, not much higher than 
the fluctuation in the presence of strong gradients 
from the surface down. Although these differences are 
probably significant, they should not be overesti¬ 
mated, in view of the wide spread within each of the 


a 4.3429 is 10 log e where the log is to the base 10 and e is 
the base of the natural logarithms. 

b In the discussion of the spread of a given set of data, it is 
very convenient to use the terms “median” and “quartile.” 
Their meaning is as follows. If all the determinations of a cer¬ 
tain quantity are arranged in the order of increasing magni¬ 
tude, the value corresponding to the midpoint of the array is 
called the median value of the spread. The point separating 
the lowest quarter of determinations from the rest is called 
the lower quartile, and the point which separates the highest 
quarter of all determinations from the rest is called the upper 
quartile. These terms will be used occasionally in the re¬ 
mainder of the chapter. 


groups of samples discussed previously. The lower 
and upper quartiles in the group of strong downward 
refractions are 46 per cent and 36 per cent, respec¬ 
tively, while the quartiles for the isothermal group are 
61 per cent and 44 per cent. It is probably justifiable 
to say that, on the average, the amplitude fluctuation 
in isothermal water is significantly higher than the 
amplitude fluctuation in the presence of strong down¬ 
ward refraction. The width of the quartile spread 
shows that even under similar conditions the magni¬ 
tude of the fluctuation itself fluctuates from sample 
to sample. In view of the large number of signals 
making up a sample, usually between 50 and 200, this 
variability is not to be explained as sampling error 
but represents an actual change in the transmission 
conditions as they affect signal fluctuation. The high 
degree of variability of fluctuation is an indication 
of the complexity of the underlying mechanism or 
mechanisms as well. 

Some information is available concerning the de¬ 
pendence of the relative standard deviation of the 
amplitude on frequency. One set of experiments, 
carried out at UCDWR, involved the simultaneous 
transmission of signals at two supersonic frequencies. 3 
The frequency pairs used were 14 and 24 kc, 16 and 
24 kc, 24 and 56 kc, and 24 and 60 kc. In 17 runs one 
frequency was either 14 or 16 kc, while the other was 
24 kc. It was found that the mean of the relative 
standard deviations at the lower frequency (14 or 
16 kc) was 38.8 per cent and at 24 kq 37.7 per cent. 
The difference is well within the root-mean-square 
spread, and is thus not significant. For the individual 
samples themselves, the difference between the fluc¬ 
tuations at the two frequencies is considerable for 
some runs, amounting to 19.2 per cent in one case. 
The root mean square difference is 8.5 per cent. As a 
result, it may be concluded that the average fluctua¬ 
tion is the same at 15 and at 24 kc, but that for any 
individual run the fluctuation may be considerably 
different at these two frequencies. In the majority of 
cases, however, high fluctuation at one frequency is 
associated with high fluctuation at the other, and 
unusually small fluctuation at one frequency tends 
to be associated with small fluctuation at the other. 
An analysis of the runs carried out with the frequency 
pairs 24 and 56 kc, and 24 and 60 kc, leads to similar 
conclusions for these frequencies. 0 

c The correlation coefficient between the magnitude of the 
fluctuation for the frequency pair 14, 16 to 24 kc was found 
to be 0.65 and for the frequency pair 24 and 56 or 60 kc, 0.68. For 
a definition of the coefficient of correlation, see Section 7.2.3. 





160 


INTENSITY FLUCTUATIONS 


At frequencies below 10 kc, it was found both at 
San Diego and at the New London laboratory of 
CUDWR that the magnitude of the fluctuation de¬ 
creases with frequency. No quantitative data are 
available. A particularly interesting result was ob¬ 
tained in a single transmission run at 5 kc at San 
Diego. It was found that at moderately short range 
the relative standard deviation of the amplitude was 
47 per cent, while beyond the computed last maxi¬ 
mum of the image interference pattern (see Chapter 
5) it dropped to 10 per cent. 



0 0.5 l.o 

SIGNAL AMPLITUDE IN ARBITRARY UNITS 

Figure 1 . Cumulative distribution function of four 

signals. 

Complete evidence is not available concerning the 
dependence of the magnitude of fluctuation on range. 
It is known that at distances of a few feet fluctuation 
of transmitted sound is negligible. From 100 yd out 
to very long ranges the average magnitude of the 
fluctuation appears to be the same at all ranges. No 
analyses have been made comparing the magnitude 
of fluctuation at different ranges under identical 
thermal conditions. 

There have been recent experiments at UCDWR 
designed to determine the possible dependence of 
fluctuation magnitudes on the depths of the projector 
and receiver. In these experiments, a cable transducer 
was used as a projector which could be lowered to 
various depths up to 300 ft. When the projector depth 
was kept constant at 16 ft, the magnitude of the 


fluctuation was found to be independent of hydro¬ 
phone depth (except for MIKE patterns). 2 When 
both the projector and receiver are deep, it is possible 
to distinguish between the direct and the surface-re¬ 
flected signal. Two runs were carried out with the 
projector at a depth of 140 ft and the hydrophone at 
a depth of 300 ft, and the direct and surface-reflected 
signals were analyzed separately. For the direct 
signal, the relative standard deviation of the ampli¬ 
tude was 9.8 per cent for the first run and 6.8 per cent 
for the second run; while for the surface-reflected 
signal the two fluctuations were 57 per cent and 51 
per cent respectively. With both projector and hydro¬ 
phone at a depth of 300 ft, the fluctuation of the 
direct signal amplitude was 6.0 per cent and of the 
surface-reflected signal 50.5 per cent. These results 
indicate that much if not most of the observed fluc¬ 
tuation is caused by mechanisms operating at or near 
the sea surface. The remaining fluctuation is proba¬ 
bly caused at least in part by the slight directivities 
of the cable-mounted projector and receiver used in 
these experiments. 

7.1.2 Probability Distributions 

The probability distribution of a set is that func¬ 
tion which tells how many members of the set lie 
between two specified values. Suppose, for instance, 
that we consider a sample of signals transmitted 
consecutively over the same transmission path. After 
these samples have been rearranged in order of in¬ 
creasing amplitude, it is then easy, by mere counting, 
to say how many of those signals have amplitudes 
less than eq, how many have amplitudes less than o 2 , 
and so on. If we divide these numbers by the total 
number of members of the sample, we obtain the 
fraction of signals with amplitudes less than a as a 
function of a, say P(a). P(a ) vanishes for a = 0, 
equals unity for a — co } and increases steadily be¬ 
tween these limits. This function is called the cumu¬ 
lative or integrated distribution function. As a very 
simple case, Figure 1 shows the integrated distribu¬ 
tion of four signals with amplitudes of 0.2, 0.4, 0.5, 
and 0.7. In the theory of statistics, it is usually as¬ 
sumed that if the number of members of the set is 
increased without limit, the shape of the function 
P(a) approaches a limiting shape in better and better 
approximation. It is this limiting shape to which a 
fundamental physical significance is ascribed. If we 
assume that the limiting function can be differenti¬ 
ated, then 


















OBSERVED FLUCTUATION 


161 


P(a) = P'(a) 


dP{a) 
da 


(4) 


is the fractional density of members of the set at a. 
In other words, p(a da is the fraction of signals with 
amplitudes between a and a + da. The function p(a) 
is often called the differential distribution function. 
Analogous concepts can be formed for intensity and 
level distributions which have here been sketched for 
amplitude distributions. If we call the intensity dis¬ 
tributions Q(I) and q(I) (the capital denoting again 



Figure 2. Cumulative distribution of the amplitudes 
of 50 signals. 


the integrated and the lower-case symbol the differ¬ 
ential distribution), and likewise the level distribu¬ 
tions W(L) and w(L), then the integrated distribu¬ 
tion functions are, of course, related to each other 
very simply by the equation 

W(L) = w /20 log = P(a) = Q(I) = (5) 

since the fraction of signals with an intensity less 
than / is identical with the fraction of signals having 
an amplitude less than a if a and I are related to each 
other by means of equation (1). For the differential 
distributions it follows that 


Similarly 


( 1 ( 1 ) = 


dQ(I) 

dl 


dP(a) da 
da dl 


= p(a)— 
a 


(7) 


because the amplitude and intensity are related by 
equation (1). 

Distribution functions can be determined experi¬ 
mentally, and the limiting distribution function will 
be approximated by the experimentally found distri¬ 
bution more and more closely as the size of the sam pie 



(tV 


Figure 3. Cumulative distribution of the amplitudes 
of 287 signals. 


is increased. On the other hand, distribution func¬ 
tions can also be predicted theoretically by assuming 
that fluctuation is caused by certain assumed mech¬ 
anisms. Figures 2 and 3 show two integrated distri¬ 
bution functions which were obtained from actual 
samples. One of these samples is plotted on proba¬ 
bility paper, on which any Gaussian distribution be¬ 
comes a straight line. d 

Two theoretically predicted distribution functions 
will be discussed here. The first of these is the so- 
called Rayleigh distribution. Let us consider, as an 


w(L) 


dW(L) 

dL 


dP(a) da 
da dL 


da 

p( 0 ) rfZ’ 


since the amplitude and level are related by equation 
(2); thus, 

® (L) = <6) 


d A Gaussian distribution is one in which the density p(a) 
is given by the function 


p(a) 


1 ^ —(a—a o) 2 /25 

V 2ird 


A Gaussian distribution will usually result if a large number of 
random processes affect the value of the argument a. The two 
parameters a 0 and 8 are the average value and the standard 
deviation of a respectively. 

























































162 


INTENSITY FLUCTUATIONS 


example of Rayleigh distribution, the intensity which 
will result if a very large number of randomly located 
scatterers return a single-frequency signal to an echo¬ 
ranging transducer. This situation is significant, be¬ 
cause it is probably the most realistic model of volume 
reverberation. Each of these scatterers will return a 
weak echo, with definite amplitude but random 
phase. The resultant of all these individual echoes 
interfering with each other in a random manner will 
be the reverberation recorded. We shall not give a 
rigorous derivation of the resulting distribution func¬ 
tion but shall sketch the argument leading to it. 



Figure 4. Reverberation amplitude produced by 
many individual echoes. 


In Chapter 2, it was explained how the amplitude 
plus phase may be combined into a single quantity, 
the “complex amplitude,” which is designated by A to 
distinguish it from the real amplitude a. Obviously, 
a is the absolute value of A. In an interference prob¬ 
lem the complex amplitude of the resultant is the sum 
of the complex amplitudes of the interfering com¬ 
ponents. If we have a large number of interfering 
components with random phases, we may plot the 
individual complex amplitudes A x , A 2 , •••, A„ and 
the resultant complex amplitude A r in the complex 
A plane as illustrated in Figure 4. The direction of 
each individual component is completely random, 
while its magnitude is fixed. Obviously, the direction 
(phase) of the resultant A r will be random. As for its 
magnitude, it is well to consider at first only its 
component in one direction, say the x axis. This 


component of A r will be the algebraic sum of the 
x components of the individual complex amplitudes 
Ai, A 2 , •••. In the mathematical theory of proba¬ 
bility it is shown that the distribution function for 
the sum of a large number of random terms is usually 
a Gaussian distribution, centered in this case about 
the zero point. In other words, the probability of the 
x component of A r having a value between x and 
x -f- dx is (1 /-\/27r5-)e~ r ' 2S dx, and the combined prob¬ 
ability of haring the x component and the y compo¬ 
nent of A r in specified brackets of infinitesimal width is 

p(x,y)dxdy = *><-> ^+y\l x dy. (8) 

Ztto- 

It is convenient to introduce the polar coordinates a 
and 9 in the complex A plane; 

a 2 = x- + y-, 


y 


tan 9 = - 
x 


(9) 


Equation (8) then assumes the form 

p(a,9)ddda = —-e -(a " 2r ' ) addda- (10) 

2ird'- 

If we wish to disregard the dependence on the phase 
angle 9, we may integrate over 9 from zero to 27 t, 
with the result 


1 


p(a)da = —e 2i } ada = —e (pc/ 5 , dl. (11) 
8 2 5- 

This last expression can be simplified by the intro¬ 
duction of the average intensity 7. By definition, this 
average intensity is given by the formula 

7 = f Iq(I)dI, (12) 

J 1=0 

in which q(I) is, according to equation (11), 


q(D = (13) 

0 “ 

Carrying out the integration, we find for 7 
8 2 

I = (14) 

pc 

which means that we have for q(I) and for Q(I) 

q(I) = y- (Ilh (15) 

and 

Q(I) = 1 — e~ (I/I) , (16) 

respectively. Whenever a signal is the resultant of a 
large number of components with random phase re- 






OBSERVED FLUCTUATION 


163 


lations, then the distribution function can be pre¬ 
dicted except for one single parameter, and this 
parameter is the average intensity. The Rayleigh 
distribution differs in this respect from a Gaussian 
distribution, which contains two adjustable param¬ 
eters, the average value and the standard deviation. 

The other distribution function to be discussed 
here is the image interference distribution. It is calcu¬ 
lated on the assumption that all the fluctuation of 
transmitted signal intensity is caused by the random 
interference of the sound transmitted directly and the 
sound reflected from the surface of the sea. The ex¬ 
tent to which this assumption is justified will be dis¬ 
cussed in Section 7.2.2. Let us assume that the ampli¬ 
tude of the direct signal alone is a t , while the ampli¬ 
tude of the surface-reflected signal by itself is a 2 . If 
the phase angle between these two components is de¬ 
noted by 9, the resultant amplitude a will be given 
by the expression 

a 2 = a? -f- a 2 + 2aia 2 cos 9. (17) 

With the large-scale geometry given, the values of ai 
and a 2 will not vary significantly; but the phase dif¬ 
ference between the two paths, 9, will change ran¬ 
domly because of the action of waves and because of 
the minute changes in position of both vessels. In 
other words, while a x and a 2 will be treated as fixed 
parameters (the values of which can, however, be 
specified only if the depths of source and receiver and 
their distance are known), 9 will be assumed to take 
all values between — x and x with equal probability. 
Since the value of a does not depend on the sign of 9, 
we shall restrict ourselves to values of 9 between 
0 and + x. The probability that 9 exceeds a certain 
value 9* equals 1 — 9*/rr, that is, the cumulative 
distribution function for A satisfies the equation 


9 

P(a ) = 1 - - 

X 


(18) 


where a and 9 are related to each other by means of 
equation (17). In other words, the fraction of signals 
for which the phase angle exceeds the value 9 is 
identical with the fraction of signals with an ampli¬ 
tude less than a. By differentiating both sides of equa¬ 
tion (18) with respect to a, we obtain an equation 
for the differential distribution, p(a), 


with 

(19 

da 


P(a ) = - 


1 d9 
x da 


(19) 


_ —2a 

V—(a* - al) 2 + 2(eti + a%)a 2 - a 1 


( 20 ) 


from equation (17). We find, then, for p(a) the ex¬ 
pression 


p(a) 


2 a 

n V — (a? — afj* + 2{a\ -(- d\)a 2 - a 4 

|oi — a 2 | ^ a ^ ai + a 2 . (21) 


Outside the limits indicated, p(o) vanishes since the 
amplitude cannot be greater than a x -(- a 2 nor less 
than |ai — a 2 |. For P(a) we find, by means of the 
relationship 

P(a) = f p(a)da, (22) 

%) a = |ai— ai\ 


the expression 
P(a) = 


11 . a 2 - (a? + a\) 

- -|— arc sm--- 

2 x 2aia 2 


= - arc sm 


'a 2 — (ai — af) 2 


4aia 2 

]ai — a 2 | ^ a ^ a i -f- a 2 


(23) 


by trigonometric transformations. Both expressions 
(21) and (23) become much simpler if it is assumed 
that the reflection from the sea surface is perfect, that 
is, if Oi = a 2 . We have, then 

p(a) = - :/TT -: 0 ^ a g 2a 1} (24) 

xV 4eq — el¬ 
and 

2 a 

P(a) = - arc sin — 0 ^ a ^ 2oq. (25) 

x 2a i 

At UCDWR, some experimentally obtained cumu¬ 
lative distribution functions of transmitted signals 
have very nearly the form of equation (25), while 
others are approximated by a Rayleigh distribution. 
All the distribution functions published at UCDWR 
are plotted as integrated distributions. This has been 
done because with a limited size of the sample the 
differential distributions would be very difficult to 
compute with any degree of reliability. Integrated 
distributions are reasonably accurate in the central 
part of the curve, but the “tails” at both ends are 
necessarily based on very few experimental data. 
This is unfortunate, because the gross features of 
integrated distributions, and particularly the central 
portions, are not very sensitive to changes in the 
character of the distribution. By definition, all inte¬ 
grated distributions are functions which increase 
steadily from zero at — °° to 1 at +°°. The central 
portions of two different distribution functions will be 
determined in their gross appearance by the location 












164 


INTENSITY FLUCTUATIONS 


of the median point and by the slope with which the 
curve passes through the median point. The central 
portion of an integrated distribution function gives, 
de facto, no more information than is contained in the 
statement of two parameters of the distribution, such 
as the mean and the standard deviation. The addi¬ 
tional information which is represented by the shapes 
of the two “tails,” must frequently be discounted 
because of the small number of signals which de¬ 
termine these shapes. 

It is true that the mere existence of a tail at the 
high amplitude end permits certain conclusions al¬ 
though these conclusions are mostly negative. If 
fluctuation were brought about exclusively by the 
interference of two signals, each having a fixed ampli¬ 
tude, then there should be a cutoff at an amplitude 
equal to the algebraic sum of the amplitudes of the 
components, corresponding to constructive inter¬ 
ference. According to equation (25), for instance, 
P(a ) should reach its maximum of 1 at an amplitude 
twice eti. The fact that there is a percentage of ampli¬ 
tudes, however small, exceeding that value proves 
that interference between two signals of equal ampli¬ 
tude cannot be the only cause of fluctuation. 

The variability of fluctuation magnitudes, which 
was touched on in Section 7.1.1, is reflected in the 
variability of the observed amplitude distributions. 
Even if large samples were processed consisting of 
thousands of signals for each sample, there is every 
reason to believe that their distribution functions 
would differ appreciably. At the present stage of the 
theorjq the details of observed distribution functions 
do not lend themselves readily to theoretical inter¬ 
pretation. 

Additional plots of observed distributions can be 
found in references 1 and 2, while additional theoreti¬ 
cal distributions are discussed in a memorandum 
from HUSL. 4 

7.1.3 Rapidity of Fluctuation 

So far, we have discussed only the typical devia¬ 
tions which individual signals show from the average. 
In this section, we shall be concerned with the time 
pattern of the fluctuation. Two sequences of signals 
could have the same relative standard deviation of 
amplitude, but could differ utterly in the nature of 
their fluctuations. For example, in one sequence the 
signal amplitudes might be distributed throughout 
the sequence in random fashion, so that a small ampli¬ 
tude signal is as likely to be followed by another small 


amplitude signal as by a large amplitude signal; while 
in the other sequence, each signal amplitude might 
be only slightly different from the amplitude of the 
preceding or following signal. In the second sequence, 
the total spread of amplitudes can be just as large 
as in the first one, if a rising or falling tendency is 
maintained through a number of consecutive signals. 

The self-correlation coefficient is the mathematical 
tool by means of which the time pattern of fluctua¬ 
tion can be expressed in quantitative form. 

The Coefficient of Self-Correlation 

Let us consider a sequence of signals which are re¬ 
ceived under apparently identical conditions. It is, of 
course, conceivable that each signal is completely 
unaffected by the strength of the preceding signal; 
this would mean that the distribution function of all 
those signals which follow immediately after signals 
of intensity are identical with the distribution func¬ 
tion of all signals (without restriction). On the other 
hand, it may be found that the signals immediately 
following signals with the intensity 7i have a distri¬ 
bution function which depends on the choice of h. 
Both of these situations seem to occur in practice. If 
the signals directly following those with intensity 7i 
tend to have intensities not too much different from 
7i, it is said that, in the sequence considered, con¬ 
secutive signals have a positive correlation. 

In order to obtain some numerical measure for the 
degree of correlation in a given sequence, we shall 
compare the difference between two consecutive sig¬ 
nals with the difference between two signals picked at 
random. Focusing our attention on intensities, for 
instance (we might as well consider amplitudes or 
levels without changing the mathematics), we shall 
compare the mean squared intensity difference be¬ 
tween two signals chosen at random with the mean 
squared intensity difference between a signal and its 
immediate predecessor. We are then concerned with 
the expression 

Si =' (7„ — 7 m ) 2 - (I n - 7„- 1 ) 2 = 2(7„7„-i — T-), 

( 26 ) 

in which n and m are to be varied independently of 
each other. The expression on the right-hand side can 
be obtained as follows. We have 

(In - I m Y - '(In - In-1) 2 

= In ~ 2. I n I m + I m ~ II + 27„7„- 1 - IU. 

In this expression, all the squared term averages, 
If, If a , and 7*_i, are equal and cancel each other. The 









OBSERVED FLUCTUATION 


165 


expression I n I m equals by definition the double sum 

_ i N 

Urn = T7 2 E Urn, 

l\ ~n,m = 1 


which in turn can be written as the product of two 
single sums, 

| N i N N 

TT 0 E Urn = — E/»E /» = 7n/ m = / 2 . 

iV n,m=l J\ i n= 1 n = l 


If there is no correlation between consecutive signals, 
then the two terms (/„ — 7 m ) 2 and (/„ — 7„-i) 2 in 
equation (26) are equal, and Si vanishes. If the cor¬ 
relation between consecutive signals is positive, then 
the rms difference between two signals picked at ran¬ 
dom will be greater than the rms difference between 
two consecutive signals, and -Si will be positive. If -Si 
should turn out to be negative, that would mean that 
the average difference square between consecutive 
signals exceeds the random value; the correlation 
between consecutive signals would then be said to be 
negative. 

The quantity -Si has the dimension of an intensity 
squared. If it is desired to obtain a measure of correla¬ 
tion which is dimensionless, it appears reasonable to 
divide -Si through by the mean squared random dif¬ 
ference, 

(In - Im) 2 = 2(P - f). (27) 

For if the correlation were perfect (that is, if each 
signal pulse had the same intensity as its predecessor, 
a situation which can obviously not be realized ex¬ 
actly), this ratio would equal unity, while for nega¬ 
tive self-correlation, the ratio can be shown never to 
drop below the value — 1. Hence, it is customary to 
measure the self-correlation of consecutive signals by 
means of the quantity 


Pi 


IJn-l ~ 7 2 
7 2 - T 


(28) 


which is called the coefficient of self-correlation for 
unit step interval. In close analogy to this quantity, 
we may define the self-correlation coefficient for an 
interval of s steps, p s , by means of the expression 


P* = 


Inins ~ 7 2 

T 2 - f 


(29) 


The averaging in the first term of the numerator is to 
be carried out by averaging over all values of the 
index n while keeping the step interval s fixed. For 
s = 0, the self-correlation coefficient equals unity, by 
definition. It can be shown that for all values of s, p s 
lies between —1 (complete anticorrelation) and 1 


(complete correlation). Furthermore, p s is an even 
function of its argument s, that is: 

p s = p_ s . (30) 


A MEAN RANGE 115 YARDS 




0 2 4 6 8 10 12 14 16 

CORRELATION INTERVAL IN SECONDS 
0 8 16 24 32 40 48 56 64 

CORRELATION INTERVAL IN SIGNALS 

Figure 5. Self-correlation coefficients of two se¬ 
quences of supersonic signals. 

Figure 5 shows two self-correlation coefficients 
which were obtained at UCDWR and which were 
computed from samples at different ranges. In both 
cases, the receiving hydrophone was in the direct 
sound field. The abscissa represents the step interval, 
marked both in terms of the number of pulses s and 
in terms of the time in seconds. These two plots, 
which are typical of the others obtained, show that 
there is a marked positive self-correlation for step 
intervals of a few seconds. It appears that the longer 
the range, the longer is the step interval of positive 
correlation (that is, the slower is the fluctuation), al¬ 
though the evidence on that point is too scanty to be 
considered conclusive. 

For many of these plots, the self-correlation be¬ 
comes negative for some step interval before it drops 
down to zero. This anticorrelation has not yet been 
explained, although it is observed more often than 
not. 














































166 


INTENSITY FLUCTUATIONS 


When the sound intensity is measured well inside 
a so-called shadow zone, there is usually no self- 
correlation for step intervals even as short as one 
second. As illustrated in Chapter 4, Figure 2B, the 
amplitude, or intensity, varies so rapidly that no 
correlation can be expected between consecutive sig¬ 
nals with the usual keying intervals. However, the 
same figure illustrates another possibility for treating 
coherence in a quantitative manner. If we consider, 
instead of a sequence of signal pulses, the amplitude 
fluctuation in a continuous signal, we may define the 
self-correlation coefficient of the amplitude as a func¬ 
tion of the continuously variable interval t as follows: 


pM 



+ r)dt - A 2 


A 2 - A 2 


(31) 


where the interval of integration T must be large 
compared with the step interval r. Figure 6 shows the 
self-correlation coefficient which was found during 



0 0.1 0.2 0.3 0.4 0.5 

T IN SECONDS 

Figure 6. Self-correlation coefficient of a 10-sec sig¬ 
nal received in the shadow zone. 


one run for sound transmitted into the shadow zone. 
The appearance of this function is similar to those in 
Figure 5, except for the enormous change in the time 
scale. Figure 7 shows the self-correlation coefficient 
which has been predicted theoretically for the in¬ 
tensity of reverberation produced by a square-topped 
single-frequency signal of length t 0 . The expression 
obtained by Eckart 3 for this coefficient is as follows: 


p(j) 



0 for | r| ^ i 0 . 


(32) 


Hidden Periodicities 

In the preceding section, the coefficient of self¬ 
correlation was introduced primarily as a mathe¬ 
matical measure of the coherence of the transmitted 
signal or, in other words, as a measure of the rapidity 
of fluctuation. In addition, the self-correlation coef¬ 



-*o o t 0 

r 


Figure 7. Theoretical self-correlation coefficient of 
the intensity of reverberation from a square-topped 
signal; 


ficient provides a powerful tool for discovering “hid¬ 
den periodicities.” A hidden periodicity is essentially 
nothing but a tendency of the fluctuation pattern to 
repeat itself with a fixed period, a tendency which is 
modified by nonperiodic disturbances. Consider, for 
instance, an ordinary pendulum which is subject to 
random forces. This pendulum will be moved to carry 
out periodic swings, but the periodicity will not be 
strict since both amplitude and phase of its vibration 
are subject to random changes. But if we were to plot 
the motion of the pendulum for a long time (large 
compared with its period), we should find that the 
self-correlation coefficient will have a maximum (al¬ 
though not quite +1) for an interval equal to the 
period of the pendulum and a minimum (although 
not quite —1) for an interval equal to one-half the 
period of the pendulum. Extending the self-correla¬ 
tion analysis to longer intervals, we should find 
another minimum at 3/2 the period, again a max- 


































CAUSES OF FLUCTUATION 


167 


imum at twice the period, and so on, these con¬ 
secutive minima and maxima becoming gradually 
more shallow until the self-correlation coefficient 
effectively approaches zero. In other words, hidden 
periodicities are revealed by the location of maxima 
and minima of the self-correlation vs interval curve.® 

It was believed, at one time, that part of the ob¬ 
served signal fluctuation could be explained as train¬ 
ing errors due to the roll and pitch of the transmitting 
vessel. Self-correlation coefficients were studied pri¬ 
marily with a view toward finding the fluctuation 
periodicities which would coincide with the known 
periodicity of the vessel. Although the results of these 
studies were at first disappointing, it is possible that 
future work will lead to more positive results. 

7.1.4 Space Pattern of Fluctuation 

In order to discover to what extent the observed 
fluctuation varies in space, an analysis was made of 
the fluctuations observed in the simultaneous out¬ 
puts of two hydrophones. 2 The two hydrophones 
were kept either at the same or at two different re¬ 
corded depths. No determination of their horizontal 
separation was made, but they are believed to have 
had a horizontal separation of between 5 and 25 ft. 
Thus, both hydrophones were at about the same 
distance from the projector, which emitted 24-kc 
signals. The number of samples analyzed is too small 
to establish any quantitative law, but indications are 
that the correlation between the simultaneous out¬ 
puts tends to become weaker as the distance of the 
two hydrophones from each other or as their joint 
distance from the sound source is increased. How¬ 
ever, in the majority of samples analyzed, the correla¬ 
tion remains significant, even at the maximum verti¬ 
cal separation of the two hydrophones, which was 
300 ft. f 

e This property of the self-correlation coefficient is incor¬ 
porated in a mathematical theorem frequently quoted as 
Khintchine’s theorem, which states that the coefficient of self¬ 
correlation is the Fourier transform of the (normalized) 
squared frequency spectrum of the time sequence considered. 
If, as in the case of the pendulum, the time sequence has a 
tendency to repeat its functional pattern, its spectrum will 
have a maximum at that frequency. This peak in the spectrum 
of the time sequence may remain undiscovered if the time 
sequence is inspected directly because of the changing phase 
relations. The squared spectrum, however, contains the abso¬ 
lute values of the squared frequency amplitudes, without 
regard for phase relationships. Consequently, its Fourier 
transform, the self-correlation coefficient, reveals the “hidden 
periodicities” more clearly than the original time sequence. 

1 With one hydrophone at 16 and the other at 300-ft depth, 
the correlation coefficient was 0.34 at a range of 950 yd and 


7.2 CAUSES OF FLUCTUATION 

It has not yet been possible to develop a theory of 
fluctuation which would permit the prediction of its 
magnitude and time rate as functions of oceano¬ 
graphic or other parameters. Nevertheless, it is of 
interest to consider the various mechanisms which 
have been considered responsible for fluctuation. 
These mechanisms may be described under three 
headings: roll and pitch of the vessel, interference 
mechanisms, and thermal microstructure of the 
ocean. 

7.2.1 Roll and Pitch of Transmitting 

Vessel 

Except in very calm weather, the transmitting 
vessel is subject to considerable roll and pitch, with 
the result that the bearing of the transmitter relative 
to the target and relative to the surface of the sea is 
not constant. Because of the directivity of the trans¬ 
mitted sound beam, a change in bearing will bring 
about a change in received signal intensity if the 
change exceeds a few degrees. This change may come 
about merely because the principal beam may miss 
the target during one phase of the roll and hit it 
during another phase. A more involved hypothesis 
considers the interference between direct and surface- 
reflected sound. In the presence of a slight upward 
refraction, the direct and surface-reflected rays to the 
target leave the projector at appreciably different 
angles. The sound received at the hydrophone is the 
result of interference between these two rays. If the 
training of the projector is changed slightly, the rela¬ 
tive intensity of the two rays will also change, as the 
projector will discriminate first against one and then 
against the other. If the two rays are out of phase by 
nearly 180 degrees, the resultant change in the in¬ 
tensity distribution of the interference pattern may 
become very appreciable, even with comparatively 
minor changes in the relative intensity of the two 
component rays. 

The chief argument in favor of roll and pitch as a 
cause of signal fluctuation was that in the earlier 
studies the self-correlation coefficient seemed to indi¬ 
cate a period of fluctuation similar to the known 
period of roll of the transmitting ship. Subsequent 


0.38 at a range of 1,750 yd. The number of signals in each 
sample was 40. For a definition of correlation coefficient, see 
Section 7.2.3. 





168 


INTENSITY FLUCTUATIONS 


work has indicated, though, that the time rate of 
fluctuation is range-dependent, that is, the time rate 
decreases as the range increases, at least in the direct 
sound field. In addition, observations made when the 


Several different models of underwater sound trans¬ 
mission have been studied which involve multiple 
paths of transmission. 



Figure 8. Observed cumulative distribution suggest¬ 
ing image interference fluctuation. 


roll of the transmitting ship was less than 2 degrees 
show about the same fluctuation as other data. Thus, 
while no definitive conclusions can be drawn at the 
present time, it seems unlikely that roll and pitch are 
dominant causes of the observed fluctuation in under¬ 
water sound transmission. 


7.2.2 Interference 

If the sound signal received at the hydrophone 
were the resultant of several individual signals trans¬ 
mitted over two or more paths, any change in the 
relative phases and amplitudes would cause a change 
in received signal strength. If the properties of the 
transmission paths were subject to random variations, 
the resulting variability of the received signal would 
depend on the characteristics of these variations. 


Two Paths 

We shall consider, first, interference between the 
direct and the surface-reflected signal. If the geome¬ 
try of one or the other path could be changed ran¬ 
domly, the result should be a fluctuation in the rel¬ 
ative amplitudes or, at least, in the relative phases of 
the two interfering signals. 

Some evidence has been accumulated which indi¬ 
cates that interference between the direct signal and 
the surface-reflected signal is at least a major con¬ 
tributing cause for the observed fluctuation. Figure 
8 shows a distribution of amplitudes similar to sev¬ 
eral which were observed at UCDWR. The theoreti¬ 
cal curve corresponding to the expression (25) (with 
a i equal to x/4 times the mean amplitude a) is super¬ 
imposed on the observed points. The moderate agree¬ 
ment indicates that during the run from which this 
distribution of amplitudes was obtained, random in¬ 
terference between two equally strong signals could 
have been the principal cause of fluctuation. On the 
other hand, a large number of observed amplitude 
distributions fail to conform to the expression (25), 
suggesting that the assumed mechanism is not al¬ 
ways the principal cause of fluctuation or, at least, 
that it is frequently modified by other mechanisms. 

Another argument in support of the hypothesis 
that fluctuation is caused, in part, by interference be¬ 
tween the direct and the surface-reflected signal is 
provided by the absence of regular image interference 
patterns for most transmission runs at supersonic 
frequencies. While traces of the pattern are regularly 
observed for transmission at low sonic frequencies 
(see Section 5.2.1), they are almost never found be¬ 
yond 100 yd at frequencies exceeding 20 kc. This 
absence has usually been explained by the size of the 
irregularities of the sea surface. While at very low 
frequencies the wavelength of underwater sound is 
large compared with most of the water waves, this is 
not true for supersonic sound. Irregularities of the sea 
surface may well replace the theoretical image inter¬ 
ference pattern by an image fluctuation; this conjec¬ 
ture is supported by the fact that fluctuation at sonic 
frequencies is often markedly lower in magnitude 
than it is at supersonic frequencies. A very striking 
plot of a transmission run at 20 kc showing both 
image effect and image fluctuation has been pub- 






























CAUSES OF FLUCTUATION 


169 



-ioo 

>- 

t 90 
in 

5 80 

z 70 

uj 60 
> 

i= 50 


RANGE IN FEET 


Figure 9. Transmission run showing image interference effect. 


lished by UCDWR 6 and is reproduced in Figure 9. 
The signal level was recorded by a sound-level re¬ 
corder. It will be noted that the ranges are very 
short, extending to not more than 90 ft. The recorder 
trace shows clearly that the amplitude of the signal 
fluctuation is greatest near the minima of the regular 
interference pattern (drawn in as a theoretical curve) 
where the magnitude of the resultant would be most 
sensitive to phase shifts of the components. This rec¬ 
ord was taken in shallow harbor water, and the sur¬ 
face was undoubtedly quite smooth. Otherwise, the 
interference pattern might not have been so notice¬ 
able. 

Finally, attention is called to the experiments 
carried out with a deep transducer, which were men¬ 
tioned in Section 7.1.1. These experiments indicate 
that the fluctuation is often reduced to a fraction of 
its usual magnitude when both the sound source and 
receiving hydrophone are so deep that the direct 
signal can be separated from the surface-reflected 
signal. It is true that some fluctuation remains, even 
when interference with the surface-reflected sound 
is eliminated; this small fluctuation may be the result 
of imperfect equipment. In all cases, however, the 
fluctuation of the direct signal is reduced drastically 
when it can be separated from the surface-reflected 
signal. The fluctuation of the surface-reflected signal 
by itself is somewhat higher than the fluctuation of 
the combined signal usually observed with a shallow 
projector. 

Several Paths 

In shallow water, or even in fairly deep water with 
a transmitter of low directivity, sound will reach the 
receiving hydrophone not only over the direct path 
and through one surface reflection, but also through 
one bottom reflection, one bottom and one surface 


reflection, etc. The number of possible paths is, 
strictly speaking, infinite, and small changes in 
geometry may bring about random phase shifts be¬ 
tween the different arrivals. Nevertheless, the distri¬ 
bution cannot be expected to approach the Rayleigh 
case, because the intensity for the paths drops rapidly 
as the number of reflections is increased, both be¬ 
cause there are recurring energy losses on reflection 
and because the high-order paths are steep-angle 
paths and therefore discriminated against by the 
transducer. Only very few of the theoretically possi¬ 
ble paths of transmission will, therefore, be effective 
in contributing to the resultant signal. It has been 
found that in the presence of bottom-reflected sound 
the rapidity of fluctuation increases, as shown by the 
oscillograph trace reproduced in Figure 10. Unfortu¬ 
nately, no quantitative information is available con¬ 
cerning the decrease in the self-correlation coefficient 
due to the contribution of bottom-reflected sound. 

Many Paths 

Figure 2 shows a distribution function obtained at 
UCDWR, and superimposed on the experimental 
points is a curve representing the Rayleigh distribu¬ 
tion. The fit is good. A model of sound transmission 
was set up in an attempt to explain this observed 
approximation to Rayleigh distribution. The model 
is based on the thermal microstructure which has 
been found to exist in the ocean 7 and which is de¬ 
scribed in Chapter 5. On the basis of ray acoustics, it 
was suggested that the irregular thermal structure of 
the ocean may give rise, simultaneously, to more than 
one ray path connecting the transmitter with the re¬ 
ceiving hydrophone. It seems reasonable to assume 
that these paths will have different travel times and 
that the signals transmitted along them are, there¬ 
fore, not in phase with each other. If the phase dif- 
















170 


INTENSITY FLUCTUATIONS 


BOTTOM R=7700 YDS HYDROPHONE DEPTHS . H= 16 FT 

REFLECTION IH=I6 FT 




R= 5900 YDS NO BOTTOM REFLECTION 



—i- - — — i — — r- - ~ r r — r —— tt — ~ ir~ — ~i — T' i r 1 n r 


Figure 10. Oscillograph trace showing the effect of bottom-reflected sound. 


ferences were random and if the average number of 
paths were sufficiently great (at least five or six), the 
resulting distribution of intensities should approach 
the Rayleigh distribution very closely. 

Against this proposed mechanism two principal ob¬ 
jections have been raised: one, experimental, the 
other, theoretical. The experimental objection is 
simply that later research has revealed that the Ray¬ 
leigh distribution is only occasionally a very good fit 
to observed transmitted intensities. 

The theoretical objection concerns the phase shifts 
expected from the observed microstructure, it is pos¬ 
sible to compute the root-mean-square difference in 
acoustical path length between two alternative paths 
through the interior of the ocean on the basis of the 
average parameters of the observed microstructure. 8 
It turns out that the magnitude of this variation in 
path length is too small to produce the random phase 
shifts as required for Rayleigh distribution. This 
argument is not entirely conclusive, because the 
microstructure parameters reported in Section 5.1.3 
were obtained on a single run and have not been 
confirmed by a repetition of the experiment. 

No critical evaluation has as yet been made of the 
multiple path hypothesis on the basis of wave 
acoustics. The multiple path hypothesis is based, 
conceptually, on ray acoustics, and it may be that 
the ray concept has been stretched in this case be¬ 


yond the limits of its validity. A similar analysis for 
a different problem has been made by CUDWR. 9 

7.2.3 Lens Action of Microstructure 

If light passes through a medium with variable 
index of refraction, such as the turbulent heated air 
above a tarred road on a hot summer day, objects 
seen through this medium are often blurred. If sun¬ 
light falls on a white screen after having passed 
through such a medium, say the hot gases surround¬ 
ing an open flame, the surface of the screen is mottled, 
with bright and dark patches alternating and chang¬ 
ing rapidly as the thermal microstructure of the trans¬ 
mitting medium is varied. This random fluctuation 
in the brightness of the illuminated screen can be ex¬ 
plained by means of the lens action of patches of 
above-average and below-average velocity of propa¬ 
gation of light. A similar explanation has been sug¬ 
gested to account for part of the fluctuation of trans¬ 
mitted sound intensity in the sea. 8 This role of the 
thermal microstructure in fluctuation is quite dif¬ 
ferent from the hypothetical interference effect dis¬ 
cussed in Section 7.2.2. While the interference effect 
is based on the coexistence of several distinct paths 
through the interior of the ocean, fluctuation because 
of refraction will be produced even over a single path. 
The theoretical treatment, not reproduced here, 







































CAUSES OF FLUCTUATION 


171 


leads to the result that if the refracting properties of 
the microstructure were alone responsible for fluctua¬ 
tion, the magnitude of fluctuation should increase 
with range. At moderate ranges the magnitude of the 
fluctuation should be proportional to the 1.5th power 
of the range. Since this hypothesis is based on ray 
acoustics, the fluctuation should be independent of 
the frequency, as long as the wavelength is short 
enough for raj" acoustics to be applicable. 

The dependence of fluctuation on range predicted 
by this hypothesis has not been confirmed by obser¬ 
vations, although the variability of the magnitude 
of the fluctuation is so great that a small effect might 
not have been discovered. For that reason, some de¬ 
pendence of fluctuation on range cannot be definitely 
ruled out. The theoretical formula connecting the 
magnitude of the predicted fluctuation with the 
parameters of the microstructure appears to lead to 
a fluctuation of a magnitude much smaller than ob¬ 
served. There is, however, one feature which appears 
to suggest that refraction by microstructure is at 
least a contributing cause of the observed fluctuation. 
It was pointed out that fluctuation caused by micro¬ 
structure should be frequency-independent for a wide 
range of frequencies. In this respect it differs from 
hypotheses based on interference, since interference 
leads to fluctuation which is critically dependent on 
frequency. It has been possible to check the depend¬ 
ence of fluctuation on frequency by transmitting 
signals simultaneously at two widely separated 
frequencies and by noting the correlation between 
their instantaneous amplitudes. 3 These trials indi¬ 
cated a partial but significant correlation between 
the fluctuations at two widely separated frequencies. 

To understand the significance of this result, it is 
necessary to explain in a few words the mathematical 
meaning of the term correlation coefficient. If there 
are two time series, say K lt K 2 , • • • , and L h L 2) • • • , 
then the correlation coefficient between them is de¬ 
fined (in close analogy to the self-correlation coeffi¬ 
cient of one time series, introduced earlier in this 
chapter) as the expression 


I\ j l J-J r\ /V 1J cy - O /C\C\ 

Pk,l = -, a k = K- — A.". (33) 

This expression equals unity if there exists a rela¬ 
tionship 

L n = aK n + p, a > 0 , n = 1, 2, 3, • • • (34) 

or, in other words, if L is a linear function of K with 


a positive slope. If a < 0, p K L will equal — 1. If there 
is some tendency of large values of L to be coupled 
with large values of K, and small values of L to be 
coupled with small values of K without the existence 
of a rigorous linear relationship (34), then p KL will 
have a positive value less than I; conversely, a 
negative value of p K L (greater than — 1) will signify 
a coupling of large values of L with small values of 
K and vice versa. If p K L vanishes, then the deviations 
of individual K values from K are statistically inde¬ 
pendent of (uncorrelated with) the deviations of the 
corresponding L values from L. 

It was found that the correlation coefficient be¬ 
tween simultaneous signals at two different fre¬ 
quencies varied from 0 to 0.75 with an average of 0.3. 
In other words, while there was some tendency for 
strong 24-kc signals to be coupled with strong 56-kc 
signals, the simultaneous signal amplitudes at these 
two frequencies were far from proportional to each 
other. The same statement holds for each of the three 
other frequency pairs at which experiments were 
performed. It must be concluded that the observed 
fluctuation is caused by a combination of mecha¬ 
nisms, of which some operate independently of the 
signal frequency (refraction and roll and pitch), 
while others depend on the transmitted frequency 
(interference). 

7.2.4 Summary 

The experiments carried out with a deep sound 
source and a deep hydrophone indicate that most of 
the observed fluctuation disappears if the whole 
transmission path is more than 100 ft below the sur¬ 
face. They also show that the surface-reflected signal 
by itself (without interference from another path) 
fluctuates more strongly than the composite signal 
observed in shallow transmission, at least when the 
incidence at the surface is not glancing. Unfortu¬ 
nately, these findings are not helpful in a choice be¬ 
tween the various mechanisms which have been 
considered. 

If roll and pitch contributed significantly to fluc¬ 
tuation, its effect on a cable-supported transducer 
would be very much less noticeable than the effect 
on a transducer rigidly connected with the hull of the 
ship; but there are not yet enough data with a shal¬ 
low-cable transducer to permit any conclusions. 
Image interference fluctuation will cease to operate 
when the surface-reflected sound can be separated 
from the direct signal. Microstructure will probably 





172 


INTENSITY FLUCTUATIONS 


be very appreciably reduced below the region of 
strong thermal gradients. 

Nevertheless, the composite evidence indicates 
that image interference is probably the most im¬ 
portant single factor contributing to the observed 
fluctuation. There may also be interference between 
the beamlets into which the irregular surface of the 
sea breaks up the incident coherent beam. In the 
deep transducer experiments, the surface-reflected 
signal showed a very high degree of fluctuation, but 
at glancing incidence the path differences may not 
be large enough to bring about random interference. 


If all interference effects could be eliminated, total 
fluctuation would probably be cut in half. 6 

The remaining fluctuation is essentially frequency- 
independent. It may be due in part to pitch and roll 
and in part to the lens effect of microstructure. Elimi¬ 
nation of either of these effects is possible in principle, 
but would require very elaborate additions to present 
sound gear. 

* Fluctuation by interference can be effectively eliminated 
by transmitting supersonic sound in a broad frequency band. 
The width of that band should probably exceed 5 kc in order 
to obtain maximum benefits. 




Chapter 8 


EXPLOSIONS AS SOURCES OF SOUND 


8.1 INTRODUCTION 

xplosive sound differs from sinusoidal sound 
both in the intensity which can be achieved with 
it and in the fact that it consists of one or more pulses 
of extremely short duration. These characteristics 
have prompted many suggestions for the employ¬ 
ment of explosive sound in communication and echo 
ranging, few of which, however, have so far been 
utilized in practice. The survey given in this chapter 
and the next of what is known about explosive sound 
is partly designed to facilitate an understanding of 
the possibilities and limitations of explosive sound 
in such applications. The study of explosive sound 
can be useful in another way, however, in that it can 
supply valuable additions to our information about 
the nature of the sea and its bottom, and about the 
causes of many of the phenomena observed in sound 
transmission. The possibilities of explosive sound as 
a research tool have accordingly been kept in mind in 
the selection of material for these chapters. 

To understand the complex phenomena which ac¬ 
company the propagation of explosive sound in the 
sea one ought to begin by finding out as precisely as 
possible just what the explosive disturbance is like, 
originally, before it has been propagated to any ap¬ 
preciable distance. Fortunately, much has been 
learned about explosions and the pressure disturb¬ 
ances which they create in the water near them. A 
detailed survey of what is known about underwater 
explosions would require a volume in itself; however, 
an effort will be made in this chapter to summarize 
briefly those parts of our knowledge of underwater 
explosions which have a bearing on the use of ex¬ 
plosions as sources of sound. In this chapter, there¬ 
fore, we shall be concerned with the disturbance at 
comparatively short ranges from the explosion, where 
its characteristics are presumably little affected by 
the departures of sea water from the concept of a 
pure homogeneous fluid. Most of the information in 
this field has been obtained in the course of experi¬ 
ments directed toward the elucidation of the damag¬ 


ing effects due to explosions. With this information 
as background, we shall be able, in Chapter 9, to dis¬ 
cuss the propagation of explosive sound through siz¬ 
able distances in the sea where departure of the 
medium from homogeneity, effects of the bottom, 
and other factors are important. 

8.2 SEQUENCE OF EVENTS IN 
UNDERWATER EXPEOSIONS 

An explosion is a process by which, in an extremely 
short space of time, a quantity of “explosive” ma¬ 
terial is converted into gas at very high temperature 
and pressure. This conversion is due to a chemical 
reaction which converts the explosive material from 
a thermodynamically unstable state to a more stable 
one with the evolution of a great amount of heat. 
This reaction, when initiated at one point of a mass 
of explosive, propagates itself rapidly until all the 
mass is involved. The propagation may take place 
in either of two ways, called respectively burning and 
detonation. In burning, the contact of the hot gaseous 
products of the reaction with the untransformed por¬ 
tion causes a reaction to take place at the surface of 
the latter, the rate of transformation being slow 
enough so that the boundary between transformed 
and untransformed material advances with a speed 
slower than the speed with which the pressure gener¬ 
ated by the reaction is propagated through the mass. 
In detonation, on the other hand, the reaction takes 
place so rapidly that it can keep up with the pressure 
wave, which in this case is known as a detonation 
wave. These two processes, which will be discussed 
more fully later, permit explosive materials to be 
divided into two rather well-defined classes: ex¬ 
plosives which detonate, commonly called high ex¬ 
plosives, and explosives which merely burn, for which 
we shall use the term propellants since the most im¬ 
portant explosives of this type are used to propel 
projectiles from guns, or as rocket fuels. A given 
quantity of high explosive will radiate considerably 
more acoustic energy when it is set off than will a like 



173 


174 


EXPLOSIONS AS SOURCES OF SOL T XD 


amount of a propellant; for this reason nearly all the 
material to be presented in these chapters concerns 
sound generated by high explosives. 

Let us therefore consider what happens when a 
quantity of high explosive is set off under water. 
First, detonation is initiated at some point of the 
explosive; this may be done, for example, by using 



Figure 1. Pressure distribution in the water at two 
instants of time following detonation of a charge of 
high explosive. 


a detonating cap containing a small quantity (about 
a gram) of an especially sensitive explosive traversed 
by a fine wire which can be suddenly heated to 
incandescence by a current of electricity. From the 
point of initiation a detonation wave spreads out in 
all directions through the explosive with a velocity 
of several thousand meters a second. In front of the 
detonation wave the material is in exactly the same 
state as before the explosion, while behind the wave 
front it is gas at a pressure of ten to a hundred thou¬ 
sand atmospheres and a temperature of several thou¬ 
sand degrees centigrade. When the detonation front 
reaches the boundary between the explosive and the 
water, this pressure is transmitted to the water, and 
a wave of intense compression starts outward through 
the water. If the pressure were not so enormous, this 
wave would be an ordinary sound wave. However, 
because of its great amplitude, the wave differs in a 
number of ways from ordinary sound waves and is 
called instead a shock wave; it bears somewhat the 
same relation to sound waves that a large breaker on 
the beach bears to an ordinary water wave. A shock 
wave is characterized by an almost discontinuous rise 
of pressure to a high value at the front of the ad¬ 


vancing wave and travels with a speed greater than 
the normal velocity of sound. The reasons for these 
characteristics will be discussed in the following 
sections. 

The pressure in the shock wave from an explosion 
dies off fairly rapidly behind the shock front, and 
by the time the shock wave has advanced to a dis¬ 
tance of the order of ten times the radius of the origi¬ 
nal mass of explosive it has become a fairly well 
localized disturbance, advancing outward and prac¬ 
tically independent of the motion of the water and 
gas in regions nearer to the center. Figure 1 shows 
schematically how the pressure may be expected to 
vary with distance from the original site of the ex¬ 
plosion at two successive instants of time as this 
state of affairs is becoming established. 

Although in these later stages it no longer affects 
the main part of the shock wave, the motion of the 
gas-filled cavity and the water immediately around 
it is by no means unimportant. At times such as those 
shown in Figure 1 the pressure in the gas cavity, 
hereafter called the bubble, is still quite high, and the 
water around it is rushing outward with a very high 
velocity. Because of the inertia of the water, this out¬ 
ward motion continues long after the force of gas 
pressure, which initiated it, has become negligible. 
As the gas bubble expands, the pressure in it drops, 
and eventually becomes far less than the normal 
hydrostatic pressure of the surrounding water. This 
excess of pressure on the outside finally brings the 
expansion to a halt, but not until the bubble has 
reached a radius which may be several dozen times 
the initial radius of the explosive. A contraction now 
sets in, and again, because of the inertia of the water, 
the bubble overshoots its equilibrium radius and the 
contraction does not stop until a very high pressure 
has been built up in the gas bubble. Several cycles of 
this expansion and contraction may take place before 
the oscillation dies out. The period of these bubble 
oscillations is of the order of a thousand times the 
duration of the pressure which the shock wave exerts 
as it passes a particular point in the water; it is usu¬ 
ally of the order of 1/30 to 1 sec, depending on the 
size of the charge and its depth. At each contraction 
a new pressure wave is sent out into the water; these 
so-called “secondary pulses” are many times less in¬ 
tense than the shock wave, but as they have a 
duration many times longer, they may contain a 
greater amount of impulse, and a comparable though 
smaller amount of energy. A quantitative theory oi 
this phenomenon will be sketched in Section 8.0. 















SHOCK FRONTS 


175 


Most of the commonly used high explosives are re¬ 
markably similar to one another in the amount of 
energy they release per unit mass, and in the relative 
amounts of energy which go into the shock wave and 
the oscillations of the bubble. Of the total work done 
by the gas on the water in its initial expansion, about 
40 to 50 per cent remains as kinetic and potential 
energy in the oscillations of the gas bubble and sur¬ 
rounding water, part of this energy being ultimately 
converted into heat by dissipative actions in the 
neighborhood of the bubble and part being radiated 
as acoustic energy in the secondary pulses. The re¬ 
maining 50 or 60 per cent of the original energy is at 
any stage divided between energy present in the 
shock wave and energy which has been converted 
into heat by dissipative processes occurring at the 
shock front. Dissipation of the latter kind is espe¬ 
cially rapid in the early stages so that by the time 
the shock wave has advanced a distance of the order 
of ten or twenty times the original radius of the 
charge, about a quarter of the original energy has 
been dissipated into heat, and the other quarter 
continues to be radiated outward in the shock wave. 
From this time on, the dissipation is much slower, 
although not negligible. 

In the preceding discussion, the phenomena have 
been described without reference to the size of the 
charge of explosive which is used. This is possible 
because explosions of all sizes are similar. If the range 
is not too great, the intensity and form of the shock 
wave, and many of the features of the bubble oscilla¬ 
tion, can be predicted exactly for one quantity of 
explosive if they are known for another quantity of 
the same explosive substance. To give a precise state¬ 
ment of the rule by which this prediction can be 
made: suppose two experiments are carried out with 
the same explosive material, the shape of the charge 
and the position of the detonation being the same in 
both cases, but the linear dimensions of the second 
charge being /? times as great as those of the first. 
Then the rule states that if the pressure is p and the 
velocity of the water is u at a distance r from the first 
charge, at time t after the detonation starts, the same 
pressure p and velocity u will obtain at a distance /3r 
in the corresponding direction from the second 
charge, at a time (3t after the detonation starts. This 
rule can be applied to the shock wave provided that 
the range from the explosion is sufficiently short so 
that the dissipative or dispersive effects responsible 
for slowing the time of rise to maximum pressure (see 
Section 9.2.1) have not had an appreciable effect on 


the pressure-time curve. The applicability of the rule 
to the later oscillation of the bubble is more limited 
and will be discussed in Section 8.6. The physical 
basis of the rule will be taken up in Section 8.4.3. 

The phenomena which occur when a propellant 
charge is set off under water are similar to those just 
described for high explosives, with the important ex¬ 
ception that because of the comparatively slow burn¬ 
ing of the explosive, the pressure transmitted to the 
water builds up gradually over a period of time, and 
does not usually create a steep-fronted shock wave. 
Thus instead of the sort of pressure-distance graph 
shown in Figure 1, a propellant would give a graph 
more like Figure 2. The division of the disturbance 

" (-•—BOUNDARY OF GAS BUBBLE 


Si | RESIDUAL FLOW 

| AROUND BUBBLE OUTGOING^ PRESSU RE WAVE 

_ - 

DISTANCE FROM CENTER OF EXPLOSION 

Figure 2. Pressure distribution in the water a short 

time after ignition of a propellant charge. 

into an outgoing pressure pulse and a re -idual bubble 
oscillation can usually still be made, but the propor¬ 
tion of the total energy which appears in the pressure 
pulse from a propellant is much smaller than that 
which appears in the shock wave from a high ex¬ 
plosive, and the maximum of the pressure is very 
much smaller. 1 The exact characteristics of the pres¬ 
sure pulse depend upon the rate of burning of the 
charge, which varies greatly depending on the type 
of propellant and the grain size. 

The following sections discuss in greater detail the 
previously mentioned features of the disturbance due 
to a high explosive. 

8.3 SHOCK FRONTS 

The steep-fronted shock waves mentioned in Sec¬ 
tion 8.2 represent a form toward which all very in¬ 
tense pressure disturbances tend to develop. In this 
section and the following section we shall show why 
this is true and shall show that many of the charac¬ 
teristics of shock waves, such as the velocity of 
propagation of the shock front and the rate of dissi¬ 
pation of energy into heat, can be expressed as func¬ 
tions of the pressure jump, that is, the amount by 
which the pressure immediately behind the shock 
front exceeds the pressure in the undisturbed water 
in front of it. Characteristics of shock waves from 








176 


EXPLOSIONS AS SOURCES OF SOUND 


explosions which depend on other factors beside the 
pressure jump will be treated in Section 8.5. 

It is a familiar fact that the laws of acoustics are a 
limiting case of the laws of hydrodynamics for a com¬ 
pressible fluid and are valid only in the limit of very 
small amplitudes of pressure and velocity. According 
to these laws of acoustics, all pressure disturbances 
are transmitted as waves with velocity c = (dp/dp), 
the derivative being understood, in the case of all 
ordinary fluids, to relate to the change of pressure p 
with density p in an adiabatic change. For the special 
case of a plane wave traveling in the positive x direc¬ 
tion, the pressure in the acoustic approximation is 
simply a function of (a: — ct), where t is the time; any 
such wave is thus propagated forward with velocity c 
without change of shape. For a disturbance of large 
amplitude this is no longer true. The shape of the 
wave will, in general, change as it progresses. The 
way in which the changes of shape take place can be 
calculated by a method of reasoning due to Riemann, 
the mathematical form of which will be given later in 
Section 8.4.1. Here we shall be content to give a 
simple qualitative explanation of Riemann’s ideas. 


p 



sion. 

Suppose we have a plane wave of the form shown 
in Figure 3A advancing in the positive x direction. 
Let the particle velocity at x = Xi be u u and let that 
at x = Xo be u 2 . If we use acoustic theory as a first 


approximation, we have iq ~ Pi/c, u 2 ~ pi/c, where 
pi and p 2 are the pressures at x, and x 2 respectively.® 
Now imagine an observer moving with the velocity iq, 
so that to this observer the fluid at the point .ri is 
instantaneously at rest. Any small additional dis¬ 
turbance, such as the bump .1, will seem to this ob¬ 
server to be propagated forward with the velocity 
Ci = (dp dp)p =Pl . Relative to the original system of 
reference, therefore, this bump will advance with 
velocity (ci + Ui). Similarly the bump B will ad¬ 
vance with velocity (co + u 2 ). Now the fact that 
pi > P 2 implies, as shown above for the acoustic 
approximation, that iq > u 2 ; moreover, the equa¬ 
tion of state of all ordinary fluids is such that this 
fact also implies ci > c 2 . Thus (ci + tq) > (C 2 + u 2 ) 
and bump A advances faster than B, so that after a 
short interval of time the pressure pulse will have 
somewhat the form shown in Figure 3B. This il¬ 
lustrates Riemann’s result, that in an advancing 
wave the parts of higher amplitude move faster than 
those of lower amplitude. If continued long enough, 
this difference in velocity would cause the high pres¬ 
sure point A to overtake the low pressure B; how¬ 
ever, before this occurs, the curve of p against x will 
acquire a vertical, or nearly vertical, tangent at some 
intermediate point C, as shown in Figure 3C. In the 
neighborhood of this point the pressure gradient and 
velocity gradient will be very large, and it will no 
longer be permissible to neglect the effects of viscos¬ 
ity and heat conduction, which have been omitted 
from Riemann’s equations. It turns out that viscosity 
and heat conduction, by converting mechanical 
energy into heat, slow up the rate of advance of the 
high pressure regions, the amount of this slowing up 
becoming greater the larger the gradient of pressure 
and velocity. Thus a stage will eventually be reached, 
as in Figure 3D, where the steepness of the rise from 
A to E is just sufficient to keep A from overtaking E. 
This state of affairs constitutes a shock wave. Since 
in practice this limiting value of the time of rise is 
extremely short, the shock wave begins with an almost 
instantaneous rise of pressure. 

The importance of this phenomenon of Riemann’s 
in the understanding of explosive sound is not merely 
that it explains the origin of shock waves, which 
could simply be taken for granted, but that it also 
explains how the characteristics of the disturbance 
behind a shock front change with the time. This vari¬ 
ation will be discussed in Section 8.5. 

a Throughout this chapter the symbol = will be used to 
denote “is approximately equal to.” 














NONLINEAR PRESSURE WAVES AND SHOCK FRONTS 


177 


From what has been said previously it would ap¬ 
pear that any mathematical theory of the propaga¬ 
tion of a shock front would have to be based on 
hydrodynamieal equations of sufficiently complicated 
form to include the effects of viscosity and heat con¬ 
duction. Fortunately, however, a very simple analysis 
based on the laws of conservation of mass, momen¬ 
tum, and energy suffices to determine the relation 
between pressure, particle velocity, temperature, and 
similar factors, just behind the shock front, and the 
velocity of propagation of the front. A detailed ap¬ 
plication of the laws of viscosity and heat conduction 
turns out to be necessary only if we are interested in 
phenomena in the shock front itself, that is, phenom¬ 
ena taking place in the very thin layer of water within 
which the abrupt rise of pressure takes place. 

The simple analysis just mentioned, due to Rankine 
and Hugoniot, will be described at length in Section 
8.4.2. It will be shown there, among other things, 
that in ordinary fluids a negative shock is impossible, 
in other words, that a discontinuity in pressure and 
density can only be propagated toward the region of 
lower pressure, and that the velocity V with which a 
shock front advances, relative to the undisturbed 
fluid in front of it, is greater than the velocity of 
sound [the value of ( dp/dp )*] in the undisturbed fluid 
ahead of the shock front. 

The thickness of the region within which the pres¬ 
sure rises from p 0 to p i is of course determined by the 
dissipative phenomena, namely, viscosity, heat con¬ 
duction, and any other sources of dissipation, such as 
bubbles, which may be present in sea water. A precise 
mathematical treatment of these factors would be 
difficult, but orcler-of-magnitude considerations to be 
given in Section 8.4.4 indicate that close to the ex¬ 
plosion this thickness should be very small; at a 
distance where the pressure jump is 100 atmospheres, 
for example, as is the case at a range of about 1 ft 
from a Number 8 detonating cap, the shock front 
should be less than 0.001 cm thick, perhaps much 
less. In this region the thickness of the shock front 
is a function only of the magnitude of the pressure 
jump and decreases as the latter increases. It might 
be supposed that the time of rise would be connected 
with the time required for the detonation wave to 
travel across the explosive charge, but this is not the 
case unless the charge is extremely elongated, since 
the Riemann “overtaking effect” will succeed in 
making the shock front vertical before the shock 
wave has advanced an appreciable distance from the 
charge. 


With increasing distance from the charge the pres¬ 
sure amplitude becomes small, and eventually the 
overtaking effect will become negligible in comparison 
with the dissipative processes which tend to smooth 
out the abrupt rise in pressure. In homogeneous sea 
water, however, the thickness of a shock front should 
remain quite small until it has traveled a considerable 
distance. Thus, for example, it can be shown that the 
thickness of the shock front at 50 yd from the ex¬ 
plosion of a detonating cap should probably be only 
a fraction of a centimeter, corresponding to a few 
microseconds or less for the time of rise of the pres¬ 
sure at a given point. 

Experimental information on the thickness of 
shock fronts, or equivalently the time of rise of the 
pressure at a given point, must be treated with cau¬ 
tion. The measured value of time of rise can easily 
be completely falsified by inadequate frequency 
response characteristics of the hydrophone and re¬ 
cording equipment; in particular, the finite size of 
the hydrophone seems to have rather more effect on 
the time of rise than one would at first suppose. A 
very careful series of experiments has been conducted 
at NRL 2 on shock waves from detonating caps con¬ 
taining about half a gram of high explosive, at 
ranges from 1 to 30 ft. In these experiments the time 
taken for the pressure in the shock wave to rise to 
its maximum value was measured under the best 
conditions as about 0.3 microsecond (^sec) at all 
ranges, and since this was about the same as the 
estimated resolving time of the apparatus used, these 
experiments support the theoretical expectation of 
the preceding paragraph that the time of rise should 
be exceedingly minute. Other experiments supporting 
this expectation have been made at the Underwater 
Explosives Research Laboratory at A oods Hole, 3 
using j/^-lb charges and ranges of the order of 10 ft; 
however, the resolving time of the apparatus for these 
experiments was only about 4 ^sec. Measurements 
made at ranges of the order of hundreds of feet, 
however, seem to show quite an appreciable time of 
rise; 4-6 this effect, which is probably instrumental 
but may possibly be related in some way to oceano¬ 
graphic conditions, will be discussed at length in 
Section 9.2.1. 

8.4 THEORY OF NONLINEAR PRESSURE 
WAVES AND SHOCK FRONTS 

The four following sections will be devoted to a 
mathematical discussion of some of the topics which 




178 


EXPLOSIONS AS SOURCES OF SOUND 


have been treated briefly in the preceding sections. 
Although this material is essential to a complete 
understanding of explosive phenomena, the con¬ 
tinuity of the chapter will not be impaired by omis¬ 
sion of these sections provided the reader is willing 
to accept on faith those results from them which 
have already been quoted. 

A more complete account of the theory of waves of 
finite amplitude and shock waves is given in a report 
issued by the Applied Mathematics Panel, 7 and in 
textbooks on hydrodynamics. 8 


8.4.1 Riemann’s Theory of Waves 
of Finite Amplitude 

In Sections 2.1.2 and 2.1.3 the equations of motion 
of a perfect fluid were derived on the assumption that 
the amplitude of the disturbance was small, so that 
the acceleration of a particle of the fluid could be ap¬ 
proximated by the partial derivative of the velocity 
with respect to time. Since we wish in this section to 
treat disturbances for which this approximation will 
not be valid, we must start from a more exact form 
of the equations of motion. As before, let x,y,z be 
three rectangular coordinates in space, t the time, 
and u x ,u y ,u z the three components of particle velocity 
in the fluid. As shown in Section 2.1.2, the x com¬ 
ponent of the acceleration of a particle of the fluid is 


du x 

dt 


du x du x du x du x 

~f" U x T Uy — -(- U z — , 
dt dx dy dz 


(1) 


and this, when multiplied by the density p, must 
equal f x , the x component of force per unit volume. 
According to Section 2.1.3,/* is related to the distri¬ 
bution of pressure p by 


fx = — 


dp 

dx 


Thus the exact equation of motion for the x com¬ 
ponent of velocity is 

1 dp 
p dx 

and applying the same reasoning to the y and z com¬ 
ponents, we get the remaining equations of motion 


du x du x du x du x 

-f- U X “f- Uy -(- U Z -- = 

dt dx dy dz 


( 2 ) 


dUy 

+ 

dUy 

+ 

d Uy 



d Uy 

1 dp 

(3) 

— 

Ux— 

Uy— 

+ 

u 


—-- 

dt 


dx 

dy 


' dz 

P dy ’ 


du x 

+ 

du x 

+ 

du 2 



du x 

1 dp 


— 

Ux -- 

Uy— 

+ 

Uz 

, - = 


(4) 

dt 


dx 

dy 


dz 

p dz 


Let us now consider a disturbance in which the pres¬ 
sure and velocity are functions only of x and t, inde¬ 


pendent of y and z. For such a disturbance the equa¬ 
tion of continuity [equation (2) of Section 2.1.1] 
becomes 


dp d(pu x 
dt dx 
and equation (2) becomes 
du 


- 0 , 


du x 1 dp 

dt ^ * dx p dx 


( 5 ) 


(6) 


Riemann discovered that the two equations (5) and 
(6) could be put into a symmetrical and useful form 
by introducing the variable 

' ’ dp 




(7) 


where p 0 is a reference value of the density which is 
most conveniently chosen equal to the density of the 
undisturbed fluid. By using this equation and the 
abbreviation c = (dp/dp)' : , equation (5) becomes 


0 


dp dip dp dip du x 

— --f- u x — -p , 

dip dt dip dx dx 


dip dip 

77 + u x ~ 
dt dx 


dip du 


du x 


- p -i- — = ( 8 ) 


dp dx 


dx 


and equation (6) becomes 
du 
dt 


du x 1 dp dp dip 

x dx p dp dip dx 

dip 
dx 


Adding equations (8) and (9) gives 

d(ip + u x ) d(ip + u x ) 

+ ( u x + c) ---- = 0, 


dt 


dx 


(9) 


( 10 ) 


and subtracting equation (9) from equation (8) gives 
similarly 

d(ip — u x ) d(ip — u x ) 

+ (M,-- = 0. (11) 


dt 


dx 


Equation (10) states that the quantity (ip + u x ) is 
propagated in the x direction with velocity ( u x + c) 
while equation (11) states that the quantity (ip — u x ) 
is propagated with velocity (u x — c), that is, since 
ordinarily c > u x , is propagated in the negative 
x direction with velocity (c — u x ). These are Rie- 
niann’s results. 

The significance of these equations can be seen by 
considering a disturbance which is initially confined 
within the range a < x < b. For such a disturbance 
both (ip + u x ) and (ip — u x ) are initially zero below 
x = a and above x = b. The region in which (ip + u x ) 








NONLINEAR PRESSURE WAVES AND SHOCK FRONTS 


179 


is different from zero will advance toward increasing 
x while the region in which — u x ) differs from zero 
will recede in the opposite direction. Eventually 
these two regions will separate and leave between 
them an interval within which both (\p + u x ) and 
0 p — Ux) are zero so that the fluid is at rest at its 
normal density p 0 . The original disturbance has thus 
been split up into two progressive waves traveling in 
opposite directions. In the wave which travels in the 
positive x direction + u x ) is finite while (\p — u x ) 
is zero; in this wave, therefore, \p = u x and both the 
density and the particle velocity are propagated for¬ 
ward with the velocity (u x + c) = {\p + c). Since for 
all ordinary fluids both \p and c increase with in¬ 
creasing pressure, this velocity of propagation will be 
greater the greater the pressure, and any disturbance 
will ultimately develop as shown in Figure 3. After 
the wave front becomes very steep, of course, the 
basic equation (2) or (G) is no longer valid and must 
be modified to take account of the effects of viscosity 
and heat conduction, which are negligible for waves 
of more gradual profile. 

Most practical applications, such as pressure waves 
produced by explosions, involve spherical waves di¬ 
verging from a source rather than plane waves of the 
type we have been discussing. It can be shown, how¬ 
ever, that spherical waves have properties very simi¬ 
lar to those just established for plane waves in that 
the high-pressure regions travel faster than the low- 
pressure regions and tend to overtake them. This 
overtaking effect becomes slower and slower as the 
wave advances farther from its source because of 
the decreasing amplitude of the disturbance due to 
spherical spreading. For this reason, a pressure pulse 
radiating from a small source has to be extremely 
intense if it is to develop a shock front by means of 
the overtaking effect ; rough calculations 9 have shown 
that pressure pulses of the amplitudes ordinarily ob¬ 
tained from echo-ranging transducers will be only 
very slightly distorted by the overtaking effect and 
will not develop shock fronts. 3 However, in the case 
of transmissions at supersonic frequencies this slight 
distortion of the wave profile might be detectable by 
a receiver tuned to twice the original frequency. 

8.4.2 The Rankine-Hugoniot 
Theory of Shock Fronts 

We have seen in the preceding sections and Figure 3 
how any pressure wave of sufficiently large amplitude 
ultimately develops an extremely steep shock front 


within which the motion of the fluid will be strongly 
influenced by factors such as viscosity and heat con¬ 
duction which do not appear in the equations of 
motion of perfect fluids. Certain characteristics of 
such a shock front can be predicted only by a theory 
which takes account of these additional factors ex¬ 
plicitly; one such characteristic, which will be dis¬ 
cussed in Section 8.4.4, is the thickness of the region 
within which the abrupt rise in pressure takes place. 
Fortunately, however, it was discovered by Rankine 
and Hugoniot in the last century that many valuable 
conclusions could be drawn merely by applying the 
laws of conservation of mass, momentum, and energy 
to the motion of the fluid, without bothering at all 
about the details of phenomena in the shock front. 

To show how this can be done, let us consider the 
mass of fluid contained in a flat cylinder having unit 
cross-sectional area and having its end planes parallel 
to, and respectively ahead of and behind, the shock 
front. A side view of such a cylinder is shown at a 
particular time t by the full line ABCD in Figure 4, 



UNDISTURBED FLUID 
p O’l°o l U = 0 


Figure 4. Cylinder of fluid traversed by a shock 
front. 


AB and CD being projections of the end faces. At a 
time dt later, the fluid which was originally in ABCD 
will occupy the cylinder A'B'CD, shown with a 
dotted boundary. Now if the pressure variation in 
the shock wave is similar to that shown in Figure 3D, 
the pressure, density, and other factors will change 
very rapidly in a very thin region near the plane SF, 
but will change much more gradually everywhere 
else. If this is the case, we can assume the thickness 









ISO 


EXPLOSIONS AS SOURCES OF SOUND 


AD of the cylinder to be very small but still very 
much greater than the thickness of the shock front, 
that is, of the region within which the rapid rise of 
pressure takes place. It will then be legitimate to 
treat the pressure, density, and velocity as having 
constant values pi, pi, U\, over that part ABFS of the 
cylinder which lies behind the shock front. Ahead of 
the shock front, of course, the fluid is at rest with 
undisturbed values p 0 , po, of the pressure and density. 
Remembering that the cylinder has unit cross sec¬ 
tion, the mass of fluid in it may be written as 

PiAS + poSD. 

If we let the boundary of the cylinder move with the 
water, this cannot change in the course of time. Now, 
after the interval dt, shown in the figure, the mass is 

PiADS' + po &D 

and since SD — S’D = T Alt 

and A'S' — AS = (V — ih)dt 

where V is the speed with which the shock front ad¬ 
vances, the two expressions given for the mass can 
be equal only if 

Pi(I — Ui) = Pol • (12) 

Similar equations can be derived by applying, in¬ 
stead of the law of conservation of mass, the laws of 
conservation of momentum or energy. Thus, the 
change in the momentum of the cylinder of Figure 4 
in the time dt is 

PiU\(V — U\)dt 

and this must be equated to dt times the force 
(pi — p 0 ) acting on the cylinder which gives 

PiUiiV - Wi) = pi - p 0 . (13) 

For the energy equation, if we denote the internal 
energy per unit mass of the fluid in front of and be¬ 
hind the shock front respectively by « n and €i and 
remember that the moving part of the fluid has 
kinetic energy per unit mass, we can write for 
the change in the total energy of the cylinder during 
the time dt 

Pi^ei + —^(l — Ui)dt — p 0 €,il dt. 

This must be equal to the product of the pressure pi 
by the distance Uidt through which the rear boundary 
of the cylinder has been pushed. Thus, we get the 
final equation 

( w i\ 

Pil «i + — J(I - Ui) — p 0 e 0 F = piUi. (14) 


The three equations (12), (13), and (14), when 
augmented by known relations between the thermo¬ 
dynamic parameters of the fluid, can be shown to 
determine all the quantities pi,pi,Ui,F,ei in terms of 
any one of them, when p 0 and p 0 are given. The equa¬ 
tions may be put in a more explicit form as follows. 
From equation (12), 


Ml 


(pi — Po) 


V 


p i 


(15) 


Inserting this and equation (12) into equation (13), 



whence, from equation (15), 


Mi 


1 / (pi - Po)(pi - Po) 

1 PoPl 


( 10 ) 


(17) 


Finally, if we insert the expression (12) into the first 
term of (14), and the expression (15) into the right- 
hand side, 



whence, using equation (17), 

P 1 — Po Ml 

«i — e 0 — pi — 

PoPi £ 


or 


*i — to 


Pi — Po 

= p 1- 

PoPl 


— i(Pi + Po) 


(Pi - Po) 

2 




(18) 


In the discussion leading to these equations, we 
have spoken of the region behind the advancing shock 
front as the “high-pressure region,” although all the 
equations which have been written would still be 
valid if pi were less than p 0 instead of greater. How¬ 
ever, it is easy to show from the energy equation (18) 
that in ordinary fluids a “rarefactional shock wave,” 
that is, one for which pi < p 0 , cannot exist. To prove 
this, consider the pressure-volume diagrams shown 
in Figure 5. The state of the undisturbed fluid of the 
shock front is represented by the point »8 0 . If the fluid 
were gradually and adiabatically compressed to den¬ 
sity pi, it would reach the state <S'b Now, according to 
the second law of thermodynamics, any sudden com¬ 
pression to this density involving irreversible proc¬ 
esses must leave the fluid in a state which is hotter 
than *SJ, in other words, since pressure normally in- 
















NONLINEAR PRESSURE WAVES AND SHOCK FRONTS 


181 


creases with increasing temperature, the point Si 
corresponding to the state of the fluid behind the 
shock front must lie above S[, as shown. That this 
is entirely consistent with equation (18) for a com- 
pressional shock wave can be seen from the upper 
half of Figure 5. The right side of equation (18) 
represents the area of the trapezoid under the straight 
line SiS 0 , while the energy difference between and 
So is represented by the area under the adiabatic 
curve between these two points. If the adiabatic 
curve is concave upward, as it is for all normal fluids, 
the area under the trapezoid will exceed the area 
under the curve, and this is consistent with the 
known fact that Si has a higher temperature, hence 
a higher energy, than SJ. For a rarefaetional shock 
wave, on the other hand, the energy of Sj is lower 
than that, of S 0 by an amount represented by the area 
under the adiabatic curve between these two points 
in the lower half of Figure 5, and since the area of the 
trapezoid is again greater than this, equation (18) 
could not be satisfied unless Si had less energy than 
Sb Thus, a rarefaetional shock is impossible for a 
normal fluid, b as is indeed to be expected from the 
fact, proved in Section 8.4.1, that regions of high 
pressure advance faster than those of low pressure. 

It is instructive to compare the equations (16) and 
(17) with the corresponding relations of acoustical 
theory to which they reduce in the limit. To verify 
the latter statement we may note that for a disturb¬ 
ance of infinitesimal amplitude, equation (18) re¬ 
duces to the law of adiabatic compression, while 
equations (16) and (13) become respectively 0 



For disturbances of large amplitude, however, it is 


b A British report 10 questions this conclusion on the basis of 
certain theoretical calculations and of some photographs of 
rarefaetional waves. However, the computed example cited 
there of a “negative shock front” is merely a normal com- 
pressional shock when viewed in a system of reference in 
which the fluid ahead of it is at rest. Moreover, the rarefac- 
tional waves which have been photographed must be regarded 
as mere acoustic disturbances; the resolution of the photo¬ 
graphs is insufficient to distinguish a discontinuous shock 
front from a gradual pressure front which has a fairly appre¬ 
ciable thickness. 

c Throughout this chapter the symbol ~ will be used to 
denote asymptotic equality; in other words, it implies that the 
quantity on the left equals the quantity on the right plus other 
terms whose ratio to the quantity written approaches zero in 
the limiting process being considered. 


easily seen from the top half of Figure 5 that for all 
ordinary fluids the value of V given by equation (16) 
is greater than the value c 0 of (dp/dp)'- in the undis¬ 
turbed fluid. For the quantity under the radical in 
equation (16) is just 1/po times the slope of the 
straight line S\So while Co is 1/po times the slope of 
the tangent to the adiabatic curve at S 0 . By a similar 
argument it can be shown that for a fluid such as 

PRESSURE 



PRESSURE 



Figure 5. Pressure-volume diagram of the changes 

occurring in a shock front. 

water, for which »?][ and Si are very close together 
(V — Ui) is less than Ci, the value of (dp/dp) 1 im¬ 
mediately behind the shock front. These results mean 
that small disturbances created behind the shock 
front may overtake it, but that no small disturbance 
can be propagated from the shock front into the un¬ 
disturbed fluid ahead. 

In fluids such as water and air, the dissipative 
phenomena taking place in the shock front gradually 
convert the mechanical energy of a shock wave into 
heat, causing the amplitude of the wave to decrease 
as it advances by an amount additional to the 
familiar decrease due to spherical divergence. A 
sufficiently intense shock wave traversing a high ex¬ 
plosive, however, can maintain itself indefinitely be¬ 
cause of the energy supplied by the chemical con¬ 
version of the explosive into gaseous products; such 
a shock wave would constitute a detonation wave. 














182 


EXPLOSIONS AS SOURCES OF SOUND 


8.4.3 The Law of Similarity 

According to the theory outlined in Sections 8.4.1 
and 8.4.2 the disturbance created in the water by an 
explosion is uniquely determined by: 

1. The forces which the explosive gases exert on 
the water near them. 

2. The Hugoniot equations (16), (17), and (18), 
which hold across the advancing shock front. 

3. The equation of continuity [equation (2) of 
Section 2.1.1], 

4. The equations of motion (2), (3), and (4), 
which hold true to a very good approximation at all 
points of the water except points in the shock front. 

5. The thermodynamic properties of the water, 
that is, the relations, such as the equation of state, be¬ 
tween pressure, density, and energy. Moreover, from 
what has been said previously concerning the nature 
of detonation waves, it is likely that the course of 
events within the explosive material itself is deter¬ 
mined by a similar set of equations, so that factor 1 
can be derived from laws of the same type as 2, 
3, 4, and 5. Now suppose a disturbance to be 
given which satisfies all these equations, and is 
described by 

V = P(x,y,z,t) 
p = p(x,y,z,t) 
u z = u z {x,y,z,t) 
u y = u y (x,y,z,t) 
u z = u z (x,y,z,t) 

Then it can easily be verified by substitution that all 
the equations mentioned in 2, 3, and 4 outlined 
previously and the laws 5 as well, will be satisfied 
at all points except those in a thin layer at the shock 
front, by a disturbance described by p',p',u' x ,u' y ,u' z 
where 

p'(x,y,z,t ) = p(0x,0y,0z,0t) 
p'(x,y,z,t) = p(0x,0y,0z,0t) 
u ' z (x,y,z,t) = u z (0x,0y,0z,0t ) 

etc., that is, by a disturbance identical with the first 
except that the distance and time scales have been 
changed by a factor 0. A scale relationship of this 
sort may be expected to hold both in the explosive 
material and in the water, and if the linear dimension 
D' of the explosive in the second case is made equal 
to 0 times the corresponding dimension D in the first 
case, the disturbances in both the explosive and the 
water can be scaled together. 

This law of similarity relating to disturbances pro¬ 


duced by different quantities of the same explosive 
has been fairly accurately verified experimentally at 
ranges for which the peak pressure in the shock wave 
is of the order of an atmosphere and above. 6 Provided 
the shape of the explosive charge and the position at 
which the detonation is initiated are the same on the 
two scales, an appreciable departure from the simi¬ 
larity law could be caused only by failure of the equa¬ 
tions of motion (2), (3), and (4) to hold behind the 
shock front in the water, or by a failure of either the 
water or the explosive to have thermodynamic prop¬ 
erties independent of the scale of times involved. A 
phenomenon of the latter type might conceivably 
occur, for example, in an aluminized explosive, if the 
reaction of the grains of aluminum with the hot gases 
were so slow that the reaction occupied an appreciable 
part of the volume behind the detonation front. 
However, the fact that significant departures from 
the scaling law have not been observed at the ranges 
mentioned indicates that such phenomena are not 
serious. As for the possibility of failure of the basic 
assumptions to be fulfilled in the water, such a failure 
could be caused only by bubbles or other extraordi¬ 
nary dissipative mechanisms; as far as is known, the 
effect of these only becomes appreciable at very long 
ranges (see Section 9.2.1). Ordinary viscosity and 
heat conduction can be shown to have a negligible 
effect on the scale used in experimental work. It 
should be remembered, of course, that the derivation 
we have given of the similarity rule does not apply 
in the very thin region of the shock front in which the 
abrupt rise of pressure takes place; as will be shown 
in the next section, the thickness of this region does 
not ordinarily scale proportionally to the factor 0 
used previously. 

8.4.4 Theoretical Thickness of a 
Shock Front 

We have seen in the preceding sections that it is 
for many purposes unnecessary to consider the de¬ 
tails of phenomena occurring in a shock front, and 
that many useful conclusions can be drawn by think¬ 
ing of a shock front merely as a surface in the fluid 
across which the pressure and other quantities change 
discontinuously. However, these conclusions will be 
valid only if the thickness of the region in which the 
pressure rise takes place is very small; and to make 
our theoretical discussion complete we should verify 
that this is the case. Moreover, a study of the factors 



NONLINEAR PRESSURE WAVES AND SHOCK FRONTS 


183 


influencing the thickness of a shock front can be 
valuable in that it may help us to evaluate and 
interpret experiments which purport to measure the 
time of rise of the pressure at the front of a shock 
wave. 

As has been explained previously, the Riemann 
overtaking effect tends to make any pressure pulse 
develop an infinitely steep front in the course of time, 
and this tendency can be counteracted only by factors 
neglected in the equations of motion of a perfect 
fluid, on which Riemann’s analysis is based. These 
factors, of which viscosity and heat conduction are 
the most obvious, must include the dissipative phe¬ 
nomena responsible for the fact that the internal 
energy of the fluid behind the shock front, as given 
by equation (18), is greater than that which the fluid 
would have if it were compressed reversibly and 
adiabatically to the density pi. This fact provides a 
clue which we can use to get a rough estimate of the 
thickness of the shock front. For, the amount of 
energy dissipated into heat per unit time by any 
given dissipative mechanism will be dependent on 
the thickness of the shock front, in other words, de¬ 
pendent on the rapidity with which the pressure 
changes from p 0 to pi. This dissipated energy must 
be equal to the product of the mass of water crossing 
the shock front per unit time by the amount of 
energy dissipated per unit mass; this quantity being 
for all practical purposes simply the difference in 
energy between the states Si and Sj in Figure 5. As 
will be shown later, the latter quantity can be calcu¬ 
lated from equation (18) in terms of the pressure 
jump (pi — p 0 ) and the known properties of water; for 
small amplitudes it is proportional to the cube of the 
pressure jump. Since all reasonable dissipative phe¬ 
nomena create heat more rapidly the more suddenly 
they are made to take place, the greater the pressure 
jump the thinner the shock front must be in order 
to dissipate the required amount of energy. Thus if 
we can show that the shock front should be quite thin 
for a fairly weak shock wave, it must be even thinner 
for a strong one. 

We shall therefore begin by calculating the ap¬ 
proximate value of the dissipated energy for a weak 
shock wave. Referring to the upper diagram in 
Figure 5, we wish to calculate the energy difference 
(ti — ej) between the states <Si and <8j. 

Since by equation (18) the difference (ei — e 0 ) 
equals the area of the trapezoid under the line <S 0 <Si 
while the difference (ej — to) equals the area under 
the adiabatic curve from to So, we must have 


(*i — ej) = area of region between SiS 0 and adia¬ 
batic curve 

= area of segment between SjS 0 and adia¬ 
batic curve + area of triangle S^SoSi. 
Now the area of the triangle SjSoSi is equal to half 
the product of its altitude (1/po — 1 /pi) by its base 
(pi — pi). Since the latter quantity is in the limit of 
small amplitudes proportional to (e t — tj), we can 
make the area of the triangle as small as we like com¬ 
pared to («i — e{) by taking the amplitude of the 
shock wave sufficiently small. Thus, for sufficiently 
weak shock waves 

(ei — ej) ~ area of segment between S[So and adia¬ 
batic curve 



where the constant k is proportional to the curvature 
of the adiabatic curve and has the numerical value 
1.5 X 10 10 in cgs units for pure water. 

Let us now consider the mechanism by which dis¬ 
sipation of energy occurs in the shock front. For any 
assumed mechanism the rate of this dissipation can be 
calculated, at least roughly, as a function of the thick¬ 
ness of the shock front and the magnitude of the 
pressure jump. Of the two most obvious mechanisms, 
viscosity and heat conduction, the former gives much 
the greater dissipation, and accordingly we shall carry 
through the calculation only for the case of dissipa¬ 
tion by viscosity. According to the theory of viscous 
fluids, the mechanical energy converted into heat per 
unit time in a fluid having a coefficient of shear viscos¬ 
ity p is given, for one-dimensional motion such as that 
in a plane wave traveling in the x direction, by 
Dissipation per unit volume per unit time 

(20) 

by the equation of continuity. If 8 is the thickness of 
the shock front — the thickness of the region within 
which most of the change in density from the value 
Po to the value pi takes place — we have, roughly, for 
a weak shock wave, 

1 dp _ c(pi — po) 

p dt p n 8 

Multiplying equation (20) by the thickness 8 gives a 
rough value for the dissipated energy per unit time 
per unit area of the shock front: 

Dissipation per unit area per unit time 

o 2 (21) 
bo 






184 


EXPLOSIONS AS SOURCES OF SOUND 


This must be equated to the product of the expres¬ 
sion (19) by the mass of water which unit area of the 
shock front traverses in unit time, that is, since we 
are assuming the shock wave to be weak, by pic: 



By solving this for 5 

Wpo _ 2 pcpl _ (22) 

k(pi ~ Po) k(pi — p 0 ) 

With the value given above for k and the value 
p = 0.01 cgs unit characteristic of pure water at 
room temperature, equation (22) becomes 

.. . 2 X 10- 

mCm (pi - p 0 )(in gm/cc) 

= - 4 5 X 1Q3 - (23) 

(pi — Po) (in dynes/cm 2 ) 

More refined calculations of this type have been 
made 11 and indicate that, in fact, nearly all the 
calculated jump in pressure occurs within an interval 
of the thickness given by equation (23). 8 

If we are to regard the relations (22) and (23) as 
giving a valid estimate of the order of magnitude of 
the thickness of a shock front in sea water, we must 
assume three things: 

1. That the Hugoniot equation (IS) is sufficiently 
accurate. 

2. That the curvature of the adiabatic for sea 
water is roughly the same as for pure water. 

3. That the rate of dissipation of energy is of the 
same order as that due to shear viscosity. 

The last of these assumptions is known to be true 
for sinusoidal disturbances in pure water at fre¬ 
quencies from 1 to 100 me. 12 Since, according to Sec¬ 
tion 5.2.2, the absorption in sea water seems to ap¬ 
proach that in pure water at frequencies near 1 me, 
it is reasonable to expect this assumption to hold in 
the sea for values of 8 between say 0.1 and 0.001 cm, 
and perhaps for much smaller values. The second 
assumption would fail if the water contained many 
bubbles, but calculations show that this would 
happen only for an unreasonably large concentration 
of bubbles. The Hugoniot equation depends for its 
validity on 8 being very small. Although no reliable 
calculation of its range of validity has been made, it 
is not hard to show that the equation (19) derived 
from the Hugoniot equation should be correct as to 
order of magnitude for explosive waves from ordinary 
sources when the pressure amplitude (pi — p 0 ) is 


greater than about 100 atmospheres. Unfortunately, 
at 100 atmospheres amplitude the value of 5 given 
by equation (23) is 4.5 X 10 -5 cm, a value so small 
that it is conceivable that assumption (3) might fail. 
Thus, about all that can be concluded from the pre¬ 
ceding analysis is that with a typical explosive source 
the thickness of the shock front at a distance where 
the pressure amplitude is 100 atmospheres is probably 
not greater than about 0.001 cm and may be much 
smaller. 

At greater distances from the explosion, a probable 
upper bound to the thickness of the shock front can 
be set by neglecting the Riemann overtaking effect, 
which tends to make the shock front steeper, and 
imagining the pressure pulse to be propagated out¬ 
ward according to the laws of acoustics, subject to 
the same attenuating mechanisms as sinusoidal 
sound. By the method of Fourier analysis (see Sec¬ 
tion 9.2.4 and Figure 13 in Chapter 9) it can be esti¬ 
mated that to avoid inconsistency with the values 
given in Section 5.2.2 for the attenuation at high 
frequencies the thickness of the shock front should 
not exceed a fraction of a centimeter after it has 
traveled 50 yd through homogeneous sea water. A 
greater thickness could be produced only by in- 
homogeneity of the medium or by a highly nonlinear 
absorption, that is, by some mechanism which would 
be much more effective in absorbing energy from a 
disturbance of large amplitude than from a weak 
disturbance. 

8.5 STRUCTURE AND DECAY OF 
SHOCK WAVES 

When a shock wave from an explosion passes a 
given point in the water, the initial behavior of the 
pressure as a function of time consists ordinarily in 
a roughly exponential dropping off, as shown 
schematically in Figure 1. The time required for the 
pressure to fall to 1/e times its value just behind the 
shock front is of the order of 15 ^sec for a Number 6 
detonating cap and GOO Msec for a 300-lb depth charge. 
This decay time depends somewhat upon the type of 
explosive being used, and it may depend slightly 
upon the range; for any given explosive it varies as 
the cube root of the charge weight, in accordance 
with the similarity rule given in Sections 8.2 and 
8.4.3. After the pressure has fallen to Ms or Mo its 
peak value, however, the decay of pressure is much 
more gradual than would correspond to an exponen¬ 
tial law. Theories of the shock wave 13 predict a “tail” 








STRUCTURE AND DECAY OF SHOCK WAVES 


185 


in which the pressure dies off with the time t approxi¬ 
mately as t~y 5 . This law cannot of course hold in¬ 
definitely, since the momentum integral J'pdt must 
be finite; the theory of bubble motion to be discussed 
in Section 8.6 predicts that the excess pressure should 
eventually go through zero and become weakly nega¬ 
tive as the gas bubble expands. For many purposes 
this tail is unimportant, but its contribution to the 
momentum integral may exceed that of the earlier 
part of the shock wave. Detailed experimental infor¬ 
mation on the tail is almost entirely lacking since 
spurious signals due to the impact of the shock wave 
on the cables leading to the pressure gauge usually 
mask the latter part of the tail. 



cylindrical charges detonated at one end rather than 
in the center, 15 - 16 for charges surrounded by an air 
pocket, 16 - 17 and for charges which fail to detonate com¬ 
pletely because of inadequate boostering. 18 More¬ 
over, even for spherical charges detonated at the 
center, the tail of the shock wave shows a small but 
reproducible hump or shoulder, whose magnitude de¬ 
pends upon the type of explosive. 

In accordance with the theory of Section 8.4.2, it is 
to be expected that the speed of advance of a shock 
wave at great distances from the explosion will be the 
normal speed of sound, but that at close distances 
the speed of advance will be considerably greater. 
Moreover, we should not be surprised to find other 
departures from the usual laws of acoustics. The 
upper diagram of Figure 6 gives some experimental 
values of the peak pressure in the shock wave as a 
function of the distance r from an explosion and shows 
fitted to these points a theoretical curve which, 
though only approximate, should give a reasonably 
reliable extrapolation of the peak pressure for smaller 
values of r. 19 If the disturbance followed the ordinary 
laws of acoustics, the curve would be a horizontal line. 
The lower diagram of Figure 6 shows the velocity of 
the shock front as a function of distance from the 
charge, this curve being related to that of the upper 
diagram by equation (16) of Section 8.4.2. It will be 
seen that as r increases the pressure becomes more 
and more nearly inversely proportional to r, as it 
should be in the acoustic approximation. It has been 
shown theoretically, however, that even in a prac¬ 
tically ideal fluid the peak pressure in a shock wave 
is not asymptotically proportional to 1/r at large 
distances, but that instead 

Constant 

P ”“~M)T (24) 


Figure 6. Peak pressure and speed of the shock front 
near a spherical charge of cast TNT. 

Under some conditions small secondary peaks or 
fluctuations may appear in the measured pressure¬ 
time curve. These may be due to any of a variety 
of causes. Sometimes the effect is spurious, being due 
to instrumental factors — for example, diffraction of 
the pressure pulse around the hydrophone and its 
supports or shock excitation of vibrations in the 
hydrophone. 2 Irregularities genuinely present in the 
explosive wave itself have been observed, however. 
Sometimes these occur under exceptional circum¬ 
stances, such as for shots fired on the bottom, 14 for 


where n is a quantity of the same order as the initial 
radius r 0 of the explosive charge. 13 This deviation 
from acoustical laws is due to the dissipation of 
energy in the shock front. The relation (24) has been 
confirmed experimentally 2 at NRL for No. 6 detonat¬ 
ing caps at ranges of 1 and 31.3 ft. The ratio 
(rpmax) 1 ft / (rp max )31.3 ft was found in these experi¬ 
ments to be 1.31 ± 0.04, while the ratio 























































186 


EXPLOSIONS AS SOURCES OF SOUND 


is 1.32. It is not to be expected, however, that equa¬ 
tion (24) will hold true indefinitely as the range is in¬ 
creased, for its theoretical derivation assumes the rise 
in pressure at the shock front to be instantaneous, 
and neglects any dissipation of energy behind the 
shock front. At large ranges neither of these assump¬ 
tions is valid, and we should expect the decrease of 
pressure with distance to obey a law similar to the 
attenuation of sinusoidal sound (see Sections 2.5 and 
9.2.1). 

The theory just mentioned also predicts that the 
duration of the pressure in a shock wave should 
slowly but continually increase as the range increases. 
This effect is due to the fact that, according to Rie- 
mann’s theory, the more intense front part of the 
wave should travel faster than the less intense tail. 
Specifically, the theory predicts that if at large 
values of the distance r from the charge the pressure 
in the wave is approximated by an exponential 
p = Pm&xe 1 e , then the duration parameter 6 should 
be given by 


0 


constant 



(25) 


where as before /’i is of the order of the radius r 0 of the 
explosive charge. The experimental verification of 
this relation is less conclusive than for the preceding 
relation (24). Although a decided decrease in 6 has 
been observed when r is decreased below a value 
corresponding to p max = 1,000 atmospheres, 20 the ex¬ 
periments of reference 2, which covered a range from 
about 3 to 80 atmospheres, showed no measurable 
increase of duration; yet an increase of the amount 
given by the relation (25) should have been measur¬ 
able. 

Under most conditions, given complete detona¬ 
tion, the peak pressures and pressure-time curves ob¬ 
tained at a given range from a charge of a given size 
are quite accurately reproducible from shot to shot, 
the deviations of individual peak pressures from the 
mean being of the order of 2 per cent. However, for 
asymmetrical charges, such as long cylinders with 
detonation initiated at one end, both the peak pres¬ 
sure and the duration of the shock wave, when 
measured at a given distance, are different in dif¬ 
ferent directions. Experiments on long cylindrical 
charges have shown that the peak pressure is greatest 
approximately at right angles to the axis of the 
cylinder and is least on the axis off the cap end. Dif¬ 
ferences as large as 40 per cent have been observed. 15 

The duration of the shock wave varies in the op¬ 


posite sense from the peak pressure, and in fact the 
impulse f pdt contained in the early part of the shock 
wave is slightly greater off the cap end, where the 
peak pressure is least, than in any other direction. As 
one would expect, charges fired with an air cavity on 
one side also show an anisotropy. 

When a charge is fired on the sea bottom, the peak 
pressure received at a point above the charge is 
somewhat greater than it would be in the absence of 
the bottom. For sand and gravel bottoms this in¬ 
crease in peak pressure has been found to be 10 to 
15 per cent. 14 In directions near the horizontal the 
amplitude tends to become smaller, as one would 
expect from the shadowing effect of irregularities on 
the bottom. The pressure-time curves from shots 
fired on the bottom are not only likely to be rather 
irregular, as mentioned previously, but tend to be 
less consistent from shot to shot than is the case for 
explosions in free water. 

8.6 SECONDARY PRESSURE WAVES 

In Section 8.2 it was mentioned that because of the 
inertia of the water which has been pushed radially 
outward by an explosion, the gas bubble undergoes 
radial oscillations. Many features of this oscillatory 
motion can be explained by a very simple theory 
which treats the water as an incompressible fluid and 
the radial flow as spherically symmetrical. 21 Let u(r,t) 
be the radial velocity of the water, measured positive 
outward at distance r from the center and at time t. 
The volume of water which passes outward in unit 
time across the surface of a sphere of radius r is 
47rr 2 w. At any given instant the flux across any two 
concentric spheres must be the same, since otherwise 
the amount of water in the shell between these two 
spheres would be increasing or decreasing. We must 
therefore have 

4 irr-u = function of t, independent of r. 

If r b (t ) is the radius of the gas bubble at time t, we 
must have 

, dr b 

u(r b ,t) = — = r b 
at 

r 2 • 

and so u(r,t ) = — • (26) 

r- 

We are now ready to apply the principle of con¬ 
servation of energy. The energy of the water and gas 
bubble consists of three parts, kinetic energy, po¬ 
tential energy due to compression of the gas in the 
bubble, and potential energy representing work done 




SECONDARY PRESSURE WAVES 


187 



TIME IN MILLISEC 

Figure 7. Ideal radius-time curve for the gas bubble from an underwater explosion. Full curve computed assuming pres¬ 
sure of gas in bubble to be given by 

p = 0.084 atmospheres. 

Dashed curve computed assuming pressure of gas in bubble to be zero. Scales: (a) No. 8 cap at 50-foot depth; (b) 1 lb 
TNT at 50-foot depth; (c) 300 lb TNT at 50-foot depth. 


against the surrounding hydrostatic pressure, which 
we shall denote by poo, since it represents the pressure 
in the water at a great distance from the bubble at the 
same level. Consider the kinetic energy first; since the 
mass of the gas in the bubble is negligible, practically 
all the kinetic energy resides in the water, and the 
amount per unit volume is The total kinetic 

energy is thus 



^pu 2 "iirr 2 dr = 2irpr\rl 


(27) 


by equation (2G). The potential energy of the gas is a 
function of its volume, and can be represented by a 
function G(r b )\ it is a negligible fraction of the total 
energy except when the radius of the bubble is small. 
The work which has been done against the external 
pressure is represented by p m times the volume of the 
bubble, or (4/3 )irpa>rl. The sum of these three terms 


must be constant in time in the approximation we 
are using 

2tt pi'lrl + ^TTpaiil + G(r b ) = W. (28) 

The behavior of r b as a function of the time is thus 
determined by solving equation (28) for r b and 
integrating. The result can be expressed in a form 
which is independent of the amount of explosive in¬ 
volved, manifesting a similarity rule of the same 
form as that given in Section 8.2 for shock wave 
pressures. For if, as before, we let r n be the radius (or 
equivalent linear dimension) of the original charge of 
explosive, G will be r® times a function of the ratio 
r b /r 0 , and W, which represents that part of the origi¬ 
nal energy which is not dissipated or carried away by 
the shock wave, will be proportional to r%. Using 
these facts, it can be seen from equation (28) that 

















188 


EXPLOSIONS AS SOURCES OF SOUND 


the solution obtained for one quantity of explosive 
will be valid for any other quantity if the scales of 
/•(, and t are changed in proportion to r 0 . The full curve 
of Figure 7, taken from a report issued by the David 
Taylor Model Basin, 22 shows the form of the radius- 
time curve obtained from integration of equation 
(28), using a function G(r b ) comparable with that 
which would obtain for a charge of high explosive; 
while the dotted curve is the one which would result 
from equation (28) assuming the same maximum 
value of r b but setting G - 0. Several scales are given 
appropriate to several sizes of charge. The variation 
of r b with t near the minimum of the contraction is 
too rapid to show on the scale of the figure. It would 
not be worth while to show this portion in greater 
detail, however, since, as will be explained presently, 
the motion of the bubble in this stage is strongly in¬ 
fluenced by gravity and other asymmetrical factors, 
and also, although less strongly, by the finite com¬ 
pressibility of the water. These effects prevent the 
present simple theory from being even approximately 
correct near the minimum. When r b is greater than 
three or four times the minimum radius shown in 
Figure 7, G(r b ) becomes small and the motion is 
practically the same as it would be if there were no 
gas in the bubble at all, as can be seen from the 
agreement of the dotted curve with the full one. For 
this portion of the curve a change in p is equivalent 
to a change in W combined with a change in time 
scale; thus, over most of the period of the oscillation 
Figure 7 applies not only to charges of different sizes, 
but to charges at different depths, provided suitable 
time and radius scales are used; these scales can be 
deduced from equation (29) below. 

The period of the motion, in the approximation 
neglecting gas pressure, is easily deduced from equa¬ 
tion (28) and turns out to be 

to = 1.135p' ! pm ;i IF' = 1.829 p l p^r bmtix (29) 
where . / 3 W V 

^ b max l . I 

\4ttp co/ 

and is the radius of the bubble at its maximum size. 
This expression, in spite of its neglect of gas pressure 
and the other effects to be discussed later, has been 
found to agree with measured values of the period to 
within a few per cent, provided the explosion takes 
place in open water well away from bounding sur¬ 
faces, which exert a perturbing effect discussed later. 
With the same proviso, the variation of the period 
with depth and size of charge agrees with the ex¬ 
ponents in equation (29) to within the accuracy of 


measurement. The measured values of the bubble 
period are found to be reproducible from shot to shot 
to within a per cent or less. 23 ’ 24 

The simple theory just outlined ignores the com¬ 
pressibility of the water and any influences which 
may make the motion asymmetrical. The compressi¬ 
bility of the water is important near the minimum 
of the contraction, when the pressure in the bubble 
is very high. During this stage of the motion the 
compression of the water near the bubble initiates 
a pressure wave which travels outward as an acoustic 
pulse. This is known as a “secondary pulse” or 
“bubble pulse,” to distinguish it from the original 
shock wave. The acoustic energy carried away by 
this secondary pulse is usually small but may, under 
exceptionally symmetrical conditions, be apprecia¬ 
ble compared with the total energy W of the oscilla¬ 
tion. Loss of energy through this effect and through 
turbulence has the consequence that the next oscilla¬ 
tion, although conforming generally to the theory of 
the preceding paragraph, is of lower amplitude than 
the first one, since it has an energy W\ which is 
smaller than W. Energy is radiated in a similar man¬ 
ner at each succeeding contraction. 

Several causes may act to prevent the motion from 
having true spherical symmetry. Most important of 
these, because it is always present, is gravity. The 
bubble, being buoyant, tends to rise; the rate of rise 
is limited by the inertia of the water around it. The 
rise is usually rather slight during the first expansion 
of the bubble, but as the bubble contracts again the 
rise is enormously accelerated and may result in a 
large portion of the total energy W being retained as 
kinetic energy in the water at the time the radius of 
the bubble is a minimum; since this energy is not 
available to compress the gas, the minimum radius 
of the bubble will not be as small as it would be if 
gravity were absent, and the secondary pressure 
pulse will be correspondingly weaker. Another effect 
of the rapid rise is to produce turbulence in the con¬ 
tracted stages; this turbulence dissipates energy and 
is probably the most important factor in the damping 
out of the oscillations. 

This rapid rise in the contracted stages has been 
the object of many theoretical studies. 22,25-27 The 
explanation of the phenomenon rests on the fact that 
a spherical cavity moving through a fluid possesses 
an “effective inertia” equal to half the mass of the 
water it displaces. The buoyancy of the gas bubble 
causes it to acquire an ever-increasing amount of 
vertical momentum, and to conserve this momentum 




SECONDARY PRESSURE WAVES 


189 


during the contraction, when the effective mass is 
greatly decreased, the velocity of rise must increase. 

Besides gravity, other effects such as proximity to 
the free surface of the water or to the bottom can 
cause departures from spherical symmetry. The ef¬ 
fects of such surfaces become appreciable when the 
distance from the bubble to the boundary surface is a 
few times the maximum radius of the bubble, and are 
of two sorts. In the first place, the period of the oscil¬ 
lation is shortened by proximity to the free surface 
and lengthened by proximity to an unyielding sur¬ 
face; secondly, a free surface repels the bubble and 
a rigid surface attracts it. This translational motion, 
which becomes very rapid in the contracted stage, 
weakens the secondary pressure pulse for the same 
reason that the rise due to gravity does. Much 
theoretical work has been done on the period and 
migration of the bubble, 27 ’ 28 and the results are in 
generally good agreement with experiment. 25 If 
gravity can be ignored, asymmetrical motion due to 
proximity to free or rigid surfaces obeys the scaling 
law of Section 8.2, the distance from the surface 
being changed in the same ratio as other linear di¬ 
mensions. The gravity effect, however, does not scale 
in the same manner; gravity is relatively more im¬ 
portant the larger the charge and the smaller the 
external hydrostatic pressure . Most features of 
the motion as affected by gravity can be approxi¬ 
mately expressed in a form which is independent of 
the size of the charge, by using a unit of length A = 
(IT /gp)', a unit of time a/A /g, and a unit of pressure 
pgrA. 25 

From the foregoing it can be seen that the form 
and strength of the secondary pressure pulses depend 
greatly on gravity and on proximity of surface, bot¬ 
tom, or objects to the explosion. The peak pressure in 
the first bubble pulse, for example, has been measured 
at values as large as 0.25, and as low as 0.06, times 
the peak pressure in the shock wave. 23 ' 24,28 ' 29 By 
contrast, the impulse J'pdl contained in any one ol 
the secondary pulses is not very sensitive to these 
factors. The amount of impulse contained within a 
few half-widths of the main pressure peak is of the 
same order as that in a corresponding portion of the 
shock wave; however, just as was the case with the 
shock wave, this impulse is considerably less than 
the amount contained in the “tails,” which in the 
case of the secondary pulses extend to both directions 
in time. The total impulse in a secondary pulse is 
probably roughly equal to the amount which would 
be calculated from the simple theory which assumes 


the water to be incompressible. It can be shown 21 
that this impulse is 

i =v~pjr- ( 30 ) 

r 





TIME IN MILLISEC 



TIME IN MILLISEC 

Figure 8. Typical pressure-time records for the first 
bubble pulse. 


At all ordinary depths this is five or ten times as 
great as the impulse in the exponential part of the 
shock wave; however, one would expect from theory 
that the impulse in the tail of the shock wave would 
















































190 


EXPLOSIONS AS SOURCES OF SOUND 


be about half of the quantity (30). In between the 
end of the tail of the shock wave and the beginning of 
the tail of the first bubble pulse there is a long period 
during which the pressure is below normal; this period 
occupies most of the time consumed by the first 
oscillation, and the negative impulse delivered during 
it is just equal to the expression (30). For most ap¬ 
plications, however, this negative pressure and the 
tail parts of the shock and bubble pulses can be 
neglected. 

In cases where migration of the bubble is slight, 
the bubble pulses show a fairly regular rise and fall of 
pressure. When migration is rapid, irregularities are 
more apt to occur, and sometimes two or more fairly 



x 


DISTANCE ESM « EH = r 2 
OISTANCE EH = r ( 

Figure 9. Superposition of direct and surface- 
reflected pulses. 

well-separated peaks are observed. 24 It has been sug¬ 
gested that these multiple impulses may be due to 
breaking of the bubble into several separate bubbles, 
which emit distinct pressure peaks in the contracted 
stage, but which coalesce when they expand again. 
A few typical oscillograms of bubble pulses, taken 
from references 23, 24, and 29, are shown in Figure 8. 
The first bubble pulse is usually by far the strongest; 
for small charges as many as eight or ten pulses have 
been counted, but for large charges usually only two 
or three bubble pulses are measurable. For some 
reason, a charge fired on or very close to the bottom 
usually gives a very weak bubble pulse, and the 
number of measurable pulses is less than for shots 
in open water. 

A caution should be added concerning the inter¬ 
pretation of oscillograms of bubble pulses; because of 
the relatively long duration of these pulses the nega¬ 
tive pulse reflected from the free surface of the water 
will often overlap the direct pulse, making the re¬ 
corded pressure appreciably different from that due 
to the direct pulse alone. The statements given 
previously apply to the latter only. 


8.7 SURFACE REFLECTION AND 
CAVITATION 

When a hydrophone is placed in the water at some 
distance from an explosive charge, the shock wave 
and secondary pulses received are modified by reflec¬ 
tion at the free surface of the water. This reflection 
is most conveniently described by the principle of 
images, which we have encountered in another ap¬ 
plication in Section 2.6.3. This principle is applicable 
whenever the pressure amplitudes are small enough 
for the la ws of ordinary acoustics to apply. Referring 
to Figure 9, the pressure produced at any instant at 



CAVITATION 

Figure 10. Modification of a surface-reflected pulse 
by cavitation. 

a hydrophone H by an explosion or other source of 
sound at E is the sum of the pressure due at that 
instant to the direct wave EH and the pressure due 
at the same instant to the reflected wave ESH. Ac¬ 
cording to the image principle, the latter pressure is 
exactly equal to the negative of the pressure which 
would be produced in the absence of a surface by a 
source E' which is the mirror image of E in the sur¬ 
face and which has the same time variation. If z c is 
the depth of the explosion, z h the depth of the 
hydrophone, and x the horizontal range, the path 
difference between these two waves is 

*2 — r l — N/( Z e + Zh ) 2 -(- X 2 — A/( Z e — Zh ) 2 + x 2 


4 Z e Z h 

T\ + To 


(31) 


When the distance ES from the explosion to the 
portion of the surface at which the reflection takes 
place is so small that the incident pressure at S is ap¬ 
preciably in excess of one atmosphere, the simple 
















SURFACE REFLECTION AND CAVITATION 


191 


image law just stated has to he modified; for, sea 
water is apparently incapable of sustaining a tension 
of any appreciable magnitude, and cavitation will 
therefore set in at any point where the pressure be¬ 
comes negative. At short ranges, therefore, the pres¬ 
sure-time curve for a shock wave and its reflection 
usually looks like the full line in Figure 10, instead 
of following the dotted curve as it would if there 
were no cavitation. 


Experiments on explosion waves in sea water 30 
strongly suggest that the water begins to cavitate as 
soon as the pressure becomes negative, and that the 
cavitation can develop sufficiently rapidly to prevent 
negative pressures from persisting even as long as 
10 nsec. This is to be expected if there are even a few 
tiny bubbles in the water. A theory of the propaga¬ 
tion of cavitation fronts has been given by Ken- 
nard. 31 ' 32 



Chapter 9 


TRANSMISSION OF EXPLOSIVE SOUND IN TI1E SEA 


9.1 INTRODUCTION 

H aving established in Chapter 8 the nature of 
explosions as sources of sound, we are ready to 
consider the results of experiments showing how 
pulses of explosive sound are affected by transmission 
through moderate and long distances in the ocean. 
The amount of experimental data available in this 
field is scanty by comparison with that which has 
been accumulated on sinusoidal sound, and since ac¬ 
curate recording of explosive pulses is possible only 
if very careful precautions are taken, one must be 
cautious in drawing conclusions from this work. 
Nevertheless, these experiments demonstrate strik¬ 
ingly the utility of explosive sound as an aid to funda¬ 
mental research on the nature of the ocean as an 
acoustic medium. Whereas in experiments using long 
pulses of sinusoidal sound the signal received at the 
hydrophone is usually inextricably compounded out 
of directly transmitted sound, scattered sound, and 
sound reflected from the surface or bottom, the ex¬ 
tremely short duration of explosive pulses makes it 
possible, in many cases, to distinguish between the 
contributions of these different mechanisms by virtue 
of the differences in time of arrival. Another charac¬ 
teristic difference between explosive and sinusoidal 
sound is that dispersion effects, which depend upon 
the phases of the various component frequencies in 
the arriving sound, can easily be studied with a 
transient disturbance, but are practically impossible 
to measure with single-frequency sound. The disper¬ 
sion accompanying the transmission of sound through 
sea water alone, although doubtless present, is very 
minute, and has never been detected; in shallow- 
water transmission, on the other hand, dispersion 
phenomena are important and can be made to give 
useful information about the bottom. Other ad¬ 
vantages of explosive sound which are significant in 
certain types of experiments include the high in¬ 
tensity attainable and the fact that explosive sources 
are relatively easy to manipulate and can be fired 
at great depths. 


The experiments to be discussed in this chapter 
shed light on a variety of problems of sound propa¬ 
gation in the ocean. For example, the variation from 
shot to shot in the sound intensity received at a dis¬ 
tance is found in Section 9.2.5 to be much smaller 
than that which is observed for sinusoidal sound, 
especially when the path of the sound lies entirely in 
isothermal water. This suggests that most of the 
variation observed with sinusoidal sound is due to 
some sort of interference phenomenon. Another ex¬ 
ample is provided by the estimates of attenuation of 
low-frequency sound, or at least of an upper bound 
to it, given in Section 9.3.2; these estimates are made 
on the basis of experiments which include ranges up 
to hundreds of miles. Other interesting results of ex¬ 
periments with explosive sound up to the present in¬ 
clude the occurrence of a reflection coefficient near 
unity for the free surface of the ocean at directions 
of incidence surprisingly close to the horizontal (Sec¬ 
tion 9.2.1), the comparison of observed intensities 
with the predictions of ray theory (Section 9.2.2), the 
occurrence of diffraction (Section 9.2.3), the deduc¬ 
tion of details of the bottom strata from shallow- 
water experiments (Section 9.4), etc. 

Before discussing the experimental material in de¬ 
tail it will be worth while to say a few words regard¬ 
ing the technique of measuring and recording ex¬ 
plosive sounds. A systematic discussion of experi¬ 
mental techniques would be beyond the scope of this 
volume; but to enable the reader to form a balanced 
opinion on past and future experiments with ex¬ 
plosive sound, some of the pitfalls and stumbling 
blocks in this field should be pointed out. In the first 
place, if actual pressure-time curves are to be re¬ 
corded, special attention has to be given to uni¬ 
formity of the response of the hydrophone and re¬ 
cording circuit, both in amplitude and in phase, over 
a wide range of frequencies. Because of the very 
short time scale when small charges are used, trouble 
may be caused by the finite time of transit of the 
sound wave across the hydrophone, and by diffrac¬ 
tion around the hydrophone and its supports. Changes 


192 


SHORT-RANGE PROPAGATION IN DEEP WATER 


193 


in the orientation of the hydrophone sometimes have 
a surprisingly large effect on the form of the recorded 
pressure-time curve. Tiny quantities of gas occluded 
on the face of the hydrophone or included in water¬ 
proofing or insulating materials can slow up the 
response to a steep-fronted pulse, and make the be¬ 
havior of the hydrophone nonlinear. Natural reso¬ 
nances in the hydrophone can be shock-excited by a 
steep-fronted pulse, causing spurious wiggles in the 
pressure-time curve, and in some cases making the 
pulse appear to last many times longer than it 
actually does. At short ranges, where relatively in¬ 
sensitive hydrophones may be used, emf’s due to the 
impact of the pressure wave on the connecting cable 
may give spurious signals. With long cables, im¬ 
pedance matching and dielectric losses may have to 
be considered. These and many other points are dis¬ 
cussed at length in other reports. 1-6 

In the following sections we shall first consider 
propagation of explosive pulses through the water 
alone, and later, in Section 9.4, shall take up pulses 
reflected from or transmitted through the bottom. 

9.2 SHORT-RANGE PROPAGATION IN 
DEEP WATER 

9 . 2.1 Attenuation and Change in 
Form of the Pulse 

As the earlier chapters of this volume have shown, 
the most important single factor affecting the shape 
and strength of a sound pulse of given frequency 
traveling through the ocean is the variation of the 
velocity of sound from point to point, due chiefly 
to temperature gradients but produced also to some 
extent by pressure and salinity gradients. Since to a 
first approximation the velocity of sound is a func¬ 
tion simply of the depth and to this approximation 
can be calculated from bathythermograph records, 
it will be convenient to separate, as far as possible, 
those features of explosive sound propagation which 
are due to this variation of velocity with depth from 
those features which are due to other properties of 
sea water and which would be encountered even 
when the bathythermograph record indicates no ap¬ 
preciable refraction. In this section we shall consider 
the latter features, recognizing, however, that unde¬ 
tected small-scale fluctuations in the velocity of 
sound may possibly be an important factor in ac¬ 
counting for them. 

One of the most interesting features to be found in 


the measurements of explosive pulses at ranges from 
30 to 2,000 yd is that with increasing range there is 
an increase in the time required for the pressure in 
the initial pulse to rise to its peak value. At shorter 
ranges, it will be remembered, this initial pulse is a 
shock wave and its time of rise is less than the re¬ 
solving time of any measuring apparatus which has 
been used (see Section 8.3). Unfortunately, the meas¬ 
urements which have been made of the time of rise 
are not sufficiently detailed to establish the cause of 
this variation with range. Table 1 summarizes the 
experimental information to date; this information 
was taken from two NDRC reports. 7 - 8 In this table 
“time of rise” is defined as the interval between the 
first measurable increase of pressure and the maxi¬ 
mum of the pressure-time curve. “Resolving time” is 
defined as the value of time of rise which the system 
would record for an instantaneous rise in pressure in 
the water. For the first set of observations this time 
was measured directly from records of shots at close 
range; for the other two sets it was merely estimated 
from acoustical and electrical characteristics of the 
hydrophone and circuit. 

The data given in Table 1 have been chosen to 
exclude any cases where the hydrophone was in or 
near the shadow zone predicted from bathythermo¬ 
graph data. They therefore presumably represent an 
effect which occurs in the absence of large-scale re¬ 
fraction, although it is not impossible that through 
inaccuracy of the computed ray diagrams some of the 
shots at the longer ranges may have been close 
enough to the shadow zone to increase the time of 
rise by virtue of the shadow-zone effect discussed 
in Section 9.2.3 and shown in Figure 9. The 
deep-water data of reference 8, which are given in 
the table, are plotted in Figure 9 of that section, 
for comparison with similar time-of-rise data taken 
in the shadow zone. It is worth noting that in these 
experiments no marked dependence of time of rise 
on the depth of the explosion was found for those 
cases where the hydrophone was not in or near the 
shadow zone; a slight increase in time of rise with 
increasing depth was observed, but this w r as not 
significantly greater than the experimental error. 
Slightly more than half of the observations of refer¬ 
ence 8 fell within +2 nsec of the means given in 
Table 1. 

For the deep-water shots off San Diego, the varia¬ 
tion of apparent time of rise with range is approxi¬ 
mately what one would expect if the resolving time of 
the apparatus were actually about 10 Msec and if the 



Table 1. Measured times of rise of the pressure in the initial pulse from an explosion, at various ranges. 


194 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


Depth of 
hydrophone 
in feet 

o c o 

’f (M Cl Cl 

2 2 2 2 7 J I 

Cl Cl Cl Cl 

Depth of 
charge 
in feet 

o o o 

<M Ol 04 

20 

20-100 

40 

10-300 

20-400 

20-400 

20-400 

iC »o ^ iC 

CO to to tO 

c i © © 

— (M — —. 

Range 
in yards 

M CO N N 

8 2 8 2 

120-133 

240 

320 

510-644 

700-900 

1,100-1,400 

1,800-1,860 

O to to o 
— cc co 
*-> o* co to 


c 

tO 


o o 
^ co 

i § o 


iO C X N 


O c 
o X 

CO 


H 

^ * 

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oi o 
o 

CO 


a . 
O ' 


£ 


3 S 3 CS 


o W 

_■ o 

£ « 


00 tO <N lO ^ 
O 


N CO Cl N 


o< 

c3 

£ 


£ 

C 

c 

r±4 


“ § .Si ^ 

03 03 ci **•« 

c3 5: - 

'—< +3 (fj 

co T3 c5 OP r 

rt 2 be h b 

o w ^ ^ a; 


c3 

co 

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C «■ 


b£ fl 


C/3 

r- ^ 


CO 


+-> sc 

CO 03 £ 

"5t 


3 is I 

o a; c« C 


■“ — b£ 

>> 5 3 


^ G 5C 


"C i- 

o 03 
0*03 

£ X 


£ Si 

£ cc 


to 

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c5 

£ 

a 


03 

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$ Q 0= 

w ^ 03 

n '■v' 


cc g ^ - 


03 - -x -r *2 A 

X 2 -J c3 

.03 ^ 0 4h X l<_ 

^ X5 TJ O o O 












































SHORT-RANGE PROPAGATION IN DEEP WATER 


195 




CAP At 40 FT 



CAP AT 80 FT 


CAP AT 5 FT 


Figure 1. Shock wave pulses received via direct and surface-reflected paths. Source: No. 8 blasting cap at depths 
indicated. Hydrophone at depth 11 feet and range 1,100 yards. Date: Feb. 26, 1942. 




196 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


explosive pulse is subject to the same frequency-de- 
pendent attenuation law as has been observed for 
supersonic sound (see Section 5.2.2). A more precise 
comparison of theory with observation would, how¬ 
ever, require knowledge not only of the exact response 
of the hydrophone and recording system to a dis¬ 
continuous change of pressure, but also of the slight 
dependence of the sound velocity on frequency. The 
WHOI data, on the other hand, are much less under¬ 
standable. The increase in recorded time of rise from 
13 /isec at short ranges to 50 nsec at 100 ft implies an 
attenuation of the high-frequency components of the 
explosive pulse, which is orders of magnitude greater 
than that encountered for supersonic sound. The 
comparative slowness of the increase in time of rise 
at greater ranges shows that this attenuation is non¬ 
linear; and the most plausible suggestion seems to be 
that it has its origin in the coating of the hydrophone, 
rather than in the sea water (see Section 8.3). 

So far we have considered only the direct pulse. 
The surface-reflected pulse may be expected to have 
a longer time of rise, or more correctly, time of fall, 
because of the diversity of possible paths from ex¬ 
plosion to hydrophone involving reflection from vari¬ 
ous wave troughs. Whether the smearing out of the 
wave due to this effect exceeds that responsible for 
the time of rise of the direct wave will of course de¬ 
pend upon the geometry and especially upon the 
roughness of the sea. Some typical oscillograms of 
shots made in an average calm sea are shown in 
Figure 1, taken from a report by UCDWR 9 and 
from reference 8. These shots show that the rough¬ 
ness of the surface has surprisingly little effect on the 
first part of the reflected pvdse, although the tail 
shows irregularities which are probably due in part 
to nonspecular reflection. The most remarkable thing 
about these records, however, is that as grazing inci¬ 
dence is approached the effective reflection coefficient 
of the surface remains close to unity far beyond the 
point at which the crests of the waves are in the 
geometric shadow created by the troughs. Figure 2 
shows the variation of the reflected amplitude with 
angle of incidence. This quantity was estimated for 
a number of shots, including those shown in Figure 1, 
by taking the difference between the reflected peak 
and the estimated pressure in the direct pulse at the 
same time. For comparison, it may be noted that, for 
a train of surface waves traveling in the direction 
from source to receiver, half of the surface of the sea 
in the region where reflection occurs would be in 
geometric shadow from source or receiver, or both, 


if the ratio of crest-to-t rough amplitude to wave¬ 
length is about one-third the angle which the incident 
ray makes with the horizontal. Since the sea was not 
unusually calm on the day the shots were made, 
Figure 2 strongly suggests that the sea surface acts as 
a flat reflecting plane for supersonic sound even when 
only the wave troughs are in the direct sound field. 
An effect of this sort has been predicted theoretically 
for the case of sinusoidal sound. 10 



ANGLE BETWEEN INCIDENT RAY AND HORIZONTAL IN RADIANS 

Figure 2. Variation of peak amplitude of surface- 
reflected pulse with angle of incidence. Horizontal 
range, 1,100 yds; depth of hydrophone, 11 ft. 

Measurements of the time interval between the 
direct and surface-reflected pulses have been re¬ 
ported in a memorandum by UCDWR 11 and in 
reference 9, and agree with the values calculated from 
the geometrical formula (31) of Section 8.7 to within 
the accuracy of measurement of the depths of cap 
and receiver. 

In the absence of refraction, one would expect the 
peak pressure of an explosive pulse having a time 
constant 6 to be attenuated at long ranges at approxi¬ 
mately the same rate as sinusoidal sound of a suitably 
chosen frequency, this “effective frequency” being 
probably a few times smaller than 1/0. A detailed 
relationship between the attenuation of a weak ex¬ 
plosive pulse and that of sinusoidal sound could be 
worked out by the methods of Fourier analysis; how¬ 
ever, such a relationship would be considerably af¬ 
fected by dispersion, that is, by any variation of the 
velocity of sound with frequency. This is a phenome¬ 
non which must occur if there is a frequency-de¬ 
pendent attenuation. A brief discussion of attenua- 




















SHORT-RANGE PROPAGATION IN DEEP WATER 


197 


tion data from the standpoint of Fourier analysis will 
be given in Section 9.2.4. In comparing the attenua¬ 
tion of explosive sound with that of sinusoidal sound, 
however, it must be kept in mind that the mechanism 
responsible for attenuation of the peak pressure in the 
initial pulse from an explosion is somewhat different 
at short and long ranges. At long ranges one may ex¬ 
pect that linear absorption and dispersion will ac¬ 
count for the decay and change ot form of the pulse; 
while at short ranges, as explained in Section 8.3 to 
8.5, the nonlinear Riemann overtaking effect plays 
the predominant role, causing the time constant of 
the pulse to increase with time and causing an at¬ 
tenuation of the peak pressure whose magnitude is 
independent of the specific mechanism responsible 
for the dissipation of energy. The range at which the 
transition from the latter type of attenuation to the 
former takes place is probably roughly the range at 
which the attenuation given by the short-range law 
(24) of Section 8.5 equals the attenuation computed 
by Fourier analysis from the linear laws of absorp¬ 
tion and dispersion which hold for weak sinusoidal 
sound. 

Observations on the variation of peak pressure with 
range do not suffice to determine the magnitude of 
the attenuation of this quantity, or even to establish 
that it is different from zero. For example, the data of 
Figures 3 and 4 of Section 9.2, taken from UCDWR 
experiments cited in reference 8, are in good agree¬ 
ment with intensity calculations which ignore at¬ 
tenuation. It would hardly be reasonable, however, 
to assume that there was practically no attenuation 
in these experiments, since the increase of time of 
rise with increasing range indicates that dissipative 
processes were appreciable. Measurements on a larger 
scale have been made at CUDWR-NLL. 12 These 
show an attenuation of about 2 db per kyd, that is, 
slightly more than that predicted by equation (24) of 
Section 8.5 for shock waves in an ideal fluid. Because 
of the difficulty of correcting accurately for the effect 
of refraction on the intensity, and because of the pos¬ 
sibility of nonlinear behavior of the hydrophone in 
CUDWR-NLL experiments, none of these results 
can be given much weight. 

Pulses have been propagated to very long ranges 
in the strata of the ocean where the velocity of sound 
is less than at shallower or deeper depths. These will 
be discussed in Section 9.3.2. Attenuation measure¬ 
ments have been made for these pulses with the use 
of recording equipment responsive to particular bands 
of frequencies. Because of the limited frequency re¬ 


sponse of the equipment, the results cannot be inter¬ 
preted in terms of peak pressure; however, as will be 
seen in Section 9.3.2, they indicate that the low- 
frequency part of the pulse is transmitted with very 
low attenuation. 

9.2.2 Effects of Refraction 

This section and the next will be concerned with 
effects which can be correlated with the variation of 
the velocity of sound with depth, as determined from 
bathythermograph data, at ranges up to a few thou¬ 
sand yards. At these ranges few if any of the ray 
paths will cross one another. In Section 9.3, on the 
other hand, we shall consider propagation over long 
ranges in a layer of minimum sound velocity where 
many different ray paths can be found leading from 
the source to the receiver. Most of the results dis¬ 
cussed in this section and the next will be taken from 
experiments conducted by UCDWR, and described 
in references 8, 9, and 11. Similar though less detailed 
results have been obtained in England at His Ma¬ 
jesty’s Anti-Submarine Experimental Establish¬ 
ment, Fairlie. 5 

In the UCDWR experiments considerable atten¬ 
tion was devoted to the securingof bathythermograph 
data at as nearly as possible the same time as the 
firing of the shots. From these data ray paths were 
computed and graphs of predicted intensity as a func¬ 
tion of depth were prepared for various values of the 
ranges, the intensities being computed from the 
geometrical divergence of the rays by the methods 
described in Chapter 3. Figure 3 shows a typical 
comparison between computed and observed peak 
pressures for a day when the upper layers of water 
were nearly isothermal. The pressure levels are 
all plotted in decibels, that is, the abscissas are 
10 logio p max- It will be seen that in the direct zone the 
observations are in reasonable agreement with the 
ray theory but that they would not agree at all well 
with an inverse square law. It is a little surprising 
that the agreement with ray theory should be so good, 
since the theoretical intensities were computed with¬ 
out any allowance for attenuation. Particularly note¬ 
worthy is the decrease in intensity as the cap is raised 
into the shadow zone from below. The 3,600-yd points 
are all in the shadow zone. Figure 4 shows a similar 
comparison for another day. On this day conditions 
were rather variable. Of the three bathythermograph 
runs taken during the morning, one showed a very 
shallow split-beam pattern while the other two 



198 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


PEAK PRESSURE LEVEL IN DECIBELS, ARBITRARY SCALE 



CURVES PREDICTED INTENSITY 


FROM RAY THEORY, WITH 
SOURCE STRENGTH CHOSEN 
TO GIVE BEST FIT TO 1200 
YARD POINTS 



showed a weak negative gradient extending to the 
surface; in the afternoon there was a strong negative 
gradient at the surface. The ray diagram and theo¬ 
retical intensities shown for the morning shots were 
constructed from an average of the three tempera¬ 
ture-depth curves taken during the morning, and 


thus are only a rough approximation to the truth 
at any one time; however, the error should not be 
serious except near the boundary of the shadow zone. 
It will be seen that the agreement of the theoretical 
and observed intensities is again fairly good. The 
reduction of intensity in the shadow zone for this 





















































SHORT-RANGE PROPAGATION IN DEEP WATER 


199 


PEAK PRESSURE LEVEL IN DECIBELS, ARBITRARY SCALE 



O RANGE 1900 YAROS, MORNING 
CURVES PREDICTED INTENSITY FROM RAY 
THEORY, WITH SOURCE STRENGTH 
CHOSEN TO GIVE BEST FIT TO 
800 YARD POINTS 

RANGE IN YARDS FEET PER SECOND 



Figure 4. Comparison of observed peak pressures with values calculated from ray theory for a negative gradient 
extending to or almost to the surface. 


case is, as one would expect, more pronounced than 
for Figure 3. 

A particularly interesting variation of intensity 
with range is shown in Figure 5. 13 This series of shots 
was made at a single depth at various ranges, on a 


day when there was a moderate negative temperature 
gradient at the surface. The velocity-depth curve and 
ray diagrams are shown in Figure G. The hydrophone 
was placed at a depth of 54 ft, just below the knee of 
the velocity-depth curve, which comes at 48 ft. This 























































200 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


causes a peculiar irregularity in the ray diagram, as 
shown in Figure 6; in this figure rays are drawn for 
initial inclinations in 0.1° steps from 1.0° to 2.5°. 
Rays whose vertices lie below 48 ft (1.0°, 1.1°, 1.2°) 
are bent downward strongly by the strong negative 
gradient below the knee. Rays rising above 48 ft 
(1.3°, 1.4°, etc.), however, are bent downward only 
weakly by the weak negative gradient above the 
knee, and diverge more rapidly; their divergence is 


\\ 

\ \ 

\ N 


I 

ENSITY COK 

APUTEO FRO 

M INVERSE 

SQ LAW 

• \ 

♦ \ 

• 



- ___ 



• 

• 

• 




INT 

RAY 

VAL 

ENSITY CO 
THEORY W 
UE FOR SO 

MPUTED FF 
TH STAND 
JRCE STREf 

OM 

*RD-^ 

JGTH 

• 








• 

• 

• 








• 


1200 1600 2000 
RANGE IN YARDS 


Figure 5. Observed and calculated variation of peak 
pressure with range for a negative temperature gradi¬ 
ent. Hydrophone depth, 54 ft; cap depth, 100 ft; 
sound conditions as shown in Figure 6. 


further increased when they curve back down through 
the knee. The result of this “double layer effect” is a 
“hole” in the sound field immediately beyond the 
1.2-degree ray, i.e., as a sharp dip in the intensity- 
range curve, as shown in Figure 5. The theoretical 
curve for this figure was not fitted to the points, but 
was computed from the known absolute strength of 
the source. At its minimum this theoretical intensity 
is some 14 db lower than that which would be pre¬ 
dicted by the inverse square law, and it is therefore 
quite significant that the observed intensities follow 
it so closely. As the shadow boundary is approached 
the observed intensities drop markedly before the 
computed shadow boundary is reached. This might 
be due to a departure from ray theory or to a slight 
error in the assumed temperature distribution near 
the surface which would cause the computed shadow 
boundary to be too far out. While data on the time 
of rise of the pressure to its peak value (see Table 2 


on page 204) favor the latter interpretation, the sys¬ 
tematic tendency of the observed shadow boundary 
to lie closer than predicted,discussed in Section 5.4.1, 
suggests that some other cause must be found for 
this apparent discrepancy. 

9.2.3 Shadow Zones and Diffraction 

As we have seen in Figures 3, 4, and 5 of Section 
9.2.2, signals of appreciable intensity are received in 
places where no rays on the sound ray diagram 
penetrate. This phenomenon is familiar in experi¬ 
ments with sinusoidal sound and has been discussed 
in Section 5.4. This and other departures of observed 
intensities and pulse forms from those computed by 
applying ray theory to bathythermograph observa¬ 
tions may be due to any of several causes. In the first 
place, the concept of propagation of sound along ray 
paths is only approximate; a more exact application 
of acoustical theory predicts that some sound should 
penetrate into the shadow zone by diffraction, and 
that in and near the shadow zone the shape of the 
pulse should be somewhat different from its shape 
close to the source. In the second place, it is known 
that the temperature in surface layers of the ocean 
is not simply a function of the depth, but varies ap¬ 
preciably from one position to another in the same 
horizontal plane. Thus a set of rays which were really 
accurately constructed would differ in many features 
from the rays which one computes from the assump¬ 
tion that temperature is a function of depth alone. 
Thirdly, the water is not homogeneous but contains 
bubbles, fish, etc., which can scatter sound and cause 
its velocity to vary with the frequency. Finally, the 
water is not at rest; portions of it may be set in 
motion relative to the rest by waves and swells, 
by tidal or other currents, and by the motion of ships, 
fish, etc. These irregularities in velocity, although 
small in comparison with the velocity of sound, may 
easily cause appreciable alterations in the shape and 
strength of an explosive pulse at ranges of the order 
of a thousand yards. 

Unfortunately, the experimental data so far avail¬ 
able are not sufficiently complete to enable many sure 
conclusions to be drawn about the mechanisms re¬ 
sponsible for the various effects observed. However, a 
few tentative conclusions can be reached regarding 
the origin of the sound which is found in the shadow 
zone in experiments such as those of references 8 and 
9 which have been discussed in the preceding section. 
Referring to Figure 5, it will be seen that in the 



















SHORT-RANGE PROPAGATION IN DEEP WATER 


201 


SOUND VELOCITY 

IN FEET PER SECOND RANGE IN YARDS 



Figure 6. Sample velocity-depth curve and ray diagram for the day on which the data of Figures 5, 7, 8, 9, 12, 14, 15, 
and 16B were taken. 


shadow zone the intensity, defined as the square of 
the pressure at the first positive peak, decreases at a 
rate of 35 or 40 db per kyd, down to a level which is 
about 30 db below the value which would be obtained 
by extrapolating the pressures obtained at short 
ranges according to the inverse square law. Less com¬ 
plete intensity data obtained on other days give com¬ 
parable values for the decrease in intensity. This de¬ 
crease is of the same order as that which would be ex¬ 
pected for diffracted sound in an ideal medium in 
which the velocity of sound varies with depth in the 
manner shown in Figure 6. 14 However, as has been 
noted in Section 5.4, a very similar decrease is ob¬ 
served for the case of 24-kc sinusoidal sound; 15 for 
this case, however, the rapid decrease of intensity 
with increasing range ceases after the intensity has 
fallen to about 40 db below the inverse square 
extrapolation, and beyond this point the decrease in 
intensity seems once again to be described by an 
inverse square law. For this and other reasons the 
supersonic signal received at a considerable distance 
inside the shadow zone is believed to arrive there by 
some sort of scattering process, rather than by dif¬ 
fraction. It does not seem likely, however, that 
scattering contributes appreciably to the observed 
intensities of the explosive pulses plotted in Figure 5 
or in Figures 3 and 4. For the disturbance produced 
at the hydrophone by scattered sound is a superposi¬ 
tion of the disturbances produced by various scatter¬ 
ing centers, and since these have different times of 
travel, the number of scattering centers which can 
contribute to the disturbance at the hydrophone at a 


given instant increases with the length of the pulse. 
The explosive pulse is so short that one would expect 
the scattered intensity to be lower by 30 db, at the 
very least, than that from a 100-msec pulse of sinu¬ 
soidal sound having the same initial amplitude and a 
frequency of the same order as that which pre¬ 
dominates in the explosive pulse in the shadow zone. 
It is thus hard to see how the scattered intensity 
could be comparable with the shadow zone inten¬ 
sities observed in Figures 3, 4, and 5. Thus we are 
forced to consider diffraction as the mechanism by 
which an explosive pulse penetrates the shadow zone, 
at least in cases such as Figure 5 and the afternoon 
shots of Figure 4, where a true shadow zone is pro¬ 
duced by downward refraction. For a split-beam pat¬ 
tern like Figure 3, the existence of a shadow zone in 
the ray diagram is due to the fact that the assumed 
velocity-depth curve has a discontinuity in slope; 
since the true variation of velocity with depth is un¬ 
doubtedly represented by a smooth curve, it is better 
in this case to speak of a zone of low intensity, rather 
than of a shadow zone, and the argument just given 
for the occurrence of diffraction is less compelling. 

The diffraction hypothesis receives support from 
a study of the shapes of pulses received in or near the 
shadow zone. Figure 7 shows the oscillograms for 
some of the shots plotted in Figure 5. It will be seen 
that as the shadow boundary is approached, the 
direct and surface-reflected pulses merge, and that 
within the shadow zone the pulse is oscillatory. The 
time of rise to the first maximum begins to increase 
suddenly at about the position of the shadow bound- 











































202 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



SHOT 17 RANGE 1950 YARDS 


SHOT 7 RANGE 960 YARDS 


SHOT 12 RANGE 1350 YARDS 


SHOT 18 RANGE 2080 YARDS 


SHOT 


RANGE 1690 YARDS 


SHOT 


RANGE 2375 YARDS 


Figure 7. Changes in the shape of an explosive pulse on passing from the direct zone into the shadow zone. Source: 
No. 8 cap at depth 100 feet and at ranges indicated. Hydrophone at depth 54 feet. Date: Apr. 3, 1942; gain of recording 
system sometimes changed between shots. 


SHOT 16 RANGE 1910 YARDS 


SHOT 23 RANGE 2520 YARDS 




SHORT-RANGE PROPAGATION IN DEEP WATER 


203 



Curves A through C are computed from diffraction theory, assuming the explosive pulse near the source to have the form 
p = Po exp (—3 X KM S ec) shown in the curve labeled initial. The conditions assumed in the calculations are compared 
with those obtaining in the experiment in the following table: 


Curve 

Velocity 

gradient 

Depth of 
source 

Depth of 
receiver 

Horizontal 
distance 
from shadow 
boundary 

A 

f 0.1 sec -1 

50 ft 

100 ft 

500 vd 

B 

Constant { 0.1 

50 “ 

50 “ 

500 “ 

C 

(0.2 

50 “ 

50 “ 

500 “ 

Observed 

See Figure 6 

50 “ 

100 “ 

460 “ 


Figure 8. Observed and computed pressure-time curves in the shadow zone. 


ary and continues to get larger and larger the farther 
one goes into the shadow zone. These features were 
observed on all occasions when shots were made in a 
shadow zone produced by downward refraction, and 
occasional repeat shots showed that the first cycle or 
so of the oscillatory pulse observed in the shadow 
zone was quite reproducible (see Figure 16C). It is 
interesting to compare these oscillograms with theo¬ 
retical pressure-time curves for sound diffracted by 
an ideal medium which has a plane surface and in 
which the velocity of sound depends only on the 
depth. Such theoretical pressure-time curves for ex¬ 
plosive sound in the shadow zone have been com¬ 


puted by CUDWR; 14 because of mathematical com¬ 
plications in the theory, however, the theoretical cal¬ 
culations have not been made for exactly the same 
conditions as any of the shots of Figure 7. Figure 8 
shows the comparison with shot 21 of Figure 7. The 
agreement is good as regards time of rise, but the 
amplitude of the negative part of the pulse is much 
greater for the observed than the theoretical case, 
and the observed wavelength is much shorter. The 
discrepancy would probably be reduced if the calcu¬ 
lation could be carried through for a velocity distri¬ 
bution approximating the observed one more closely. 
However, it is quite possible that diffraction theories 
































204 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 









A 










/ 

/ 










T 

r 

! 










j 

^SHADOW BOU 

^COMPUTED F 
jBATHYTHERM 
|DATA 

NDARIE 

ROM 

OGRAP 

S 

/ 

v' 

0 





A 

St 

/ 



/ 

/ 

/ 

/ 



SHADC 

DETEF 

1 NTEf 

)W BOU 

*MINED 

JSITY- 

NDARIE 

FROM 

RANGE 

s s 

PLOTS 

/ / 

A / 

-F 

< 

/> 

/ O 

-• 






V* 

A ^ 

A 


y<3 > 







500 


1000 1500 2000 

RANGE IN YARDS 


-•COMPOSITE OF ALL DATA TAKEN OVER A TWO MONTH PERIOD, AVER¬ 
AGED BY RANGE GROUPS, AND WITH EXCLUSION OF ALL SHOTS MADE 
BEYOND THE SHADOW BOUNDARY AS COMPUTED FROM BATHY¬ 
THERMOGRAPH DATA 

--OSHOTS MADE AT 100 FOOT DEPTH, MORNING OF APRIL 3, 1942, WITH 
HYDROPHONE AT 54 FEET 

— Ashots made at 50 foot depth, afternoon of april 3, 1942 , 

WITH HYDROPHONE at 54 FEET 


Figure 9. Dependence of time of rise on range, show¬ 
ing influence of the shadow zone. 


observed and computed shadow boundaries are 
marked on the figure. The abrupt increase in time of 
rise on crossing the observed shadow boundary is 
quite conspicuous, and was noticed in UCDWR ex¬ 
periments on all days when strong downward refrac¬ 
tion was present. For comparison, Figure 9 also shows 
as a full curve the average values given in Table 1 
of Section 9.2.1, which at each range represent the 
time of rise when refraction conditions are such that 
the hydrophone is in the direct zone. It will be seen 
that out to the shadow boundary all values agree to 
within the fluctuations of the data. 

The sharpness of the increase in time of rise as the 
shadow boundary is crossed suggests using a plot 
like Figure 9 to determine the location of the shadow 
boundary. Table 2, taken from reference 9, gives a 
comparison of the range to the shadow boundary de¬ 
termined in this way with the range as deduced from 
the bathythermograph measurements, and also with 
the range as deduced from a plot of peak intensity 
against range, such as Figure 5. It will be seen that 
time of rise and peak pressure always give very 
nearly the same position for the shadow boundary, 
but that this position does not always agree well with 
that given by the bathythermograph. This is not sur¬ 
prising, since such things as surface waves and small 
changes of temperature very close to the surface can 


Table 2. Comparison of three methods of determining the range to the shadow boundary.* 


Date, 1942 

Depth of explosion in feet 

April 2 
100 

April 3 
50 

April 3 
100 

April 8 
50 

April 9 

50 

Range from bathythermograph (yd) 

1,800 

1,490 

1,940 

1,080 

1,010 

Range from time of rise (yd) 

1,170 

1,100 

1,430 

1,150 

1,100 

Range from peak pressure (yd) 

1,200 

1,250 

1,450 

1,170 

1,170 


* Depth of hydrophone, 54 ft in all cases. 


based on the concept of a horizontally stratified 
medium will prove inadequate to explain the experi¬ 
mental results, and that some more complicated proc¬ 
ess must be considered. 

Figure 9 shows how the time of rise to the first 
pressure maximum is affected by crossing the shadow 
boundary. The lower dashed curve is for the same 
shots as Figures 5 and 7, while the upper dot-dash 
curve is for shots at shallower depth at a different 
time on the same day. Velocities and ray diagrams 
for this day have been given in Figure (>. Since the 
shadow boundaries computed from the temperature 
data do not agree very well with the boundaries de¬ 
termined empirically from the behavior of sound in¬ 
tensity as a function of range (see Table 2) both the 


have quite an appreciable influence on the position 
of the shadow boundary, and the shadow boundary 
is determined by the distribution of temperature over 
a large area of the sea, while the bathythermograph 
measures temperatures on only one vertical line. 

Many of the oscillographic pressure-time records 
obtained of explosive pulses are much less simple and 
comprehensible than the examples which have been 
selected for discussion in the preceding paragraphs. 
Some of the irregularities can apparently be explained 
in terms of multiple ray paths, while others are more 
puzzling. Figure 10 shows some typical oscillograms 
obtained on a day when the bathythermograph 
showed that there were alternate layers of large and 
small temperature gradients, which should have pro- 








































SHORT-RANGE PROPAGATION IN DEEP WATER 


205 




SHOT II RANGE 630 YARDS 


SHOT 16 RANGE 1290 YARDS 


SHOT 18 RANGE 1370 YARDS 


SHOT 15 RANGE 1.1 10 YARDS SHOT 19 RANGE 1450 YARDS 


Figure 10. Records showing multiple path effects. Source: No. 8 cap at depth 100 feet and at ranges indicated. 
Hydrophone at depth 54 feet. Date: Apr. 2, 1942. Gain of the recording system was sometimes changed between shots. 


duced considerable crossing of the ray paths. 9 The 
630-yd oscillogram has very much the form one would 
expect if there were two ray paths leading from cap 
to hydrophone. That for 800 yd suggests a number 
of ray paths, but the arrivals are less sharp. It is diffi¬ 


cult to correlate these features in detail with the 
calculated ray diagram, however, and there appear 
to be variations which make it hard to pick out sys¬ 
tematic trends in the characters of the oscillograms 
as the range is gradually increased or decreased. 






206 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


SOUNO VELOCITY 
RELATIVE TO MAXIMUM 

RANGE IN YARDS IN FEET PER SECOND 



The three oscillograms on the right of Figure 10 do 
seem to show a regular trend, however, in that with 
increasing range the first pulse becomes rapidly 
weaker in comparison with the second, and its time 
of rise increases. Comparison with the oscillograms of 
Figure 7, which show the same trends, suggests that 
the first pulse may have reached the hydrophone by 
diffraction, while the second corresponds to a direct 
ray. The possibility of a phenomenon of this sort 
is suggested by calculations which have been made 
on ray paths for this day, a few of which are shown 
in Figure 11. The ray ABC crosses the 100-ft depth 
line at a greater value of the horizontal range than do 
rays of slightly greater or slightly smaller inclina¬ 
tions so that this ray and its neighbors have an en¬ 
velope, or caustic, passing through B. This caustic 
forms a shadow boundary as far as rays of inclina¬ 
tions near that of ABC are concerned, 16 although it 
happens in this case that rays having considerably 
greater inclinations fall outside the caustic. The com¬ 
plete ray diagram would of course be quite compli¬ 
cated, and attempts at a detailed correlation of the 
oscillograms with the bathythermograph data have 
not been very successful. 

The oscillograms shown in Figure 12 are for shots 
made on the same day as those in Figure 7; bathy¬ 
thermograph data and ray diagrams for this day have 
been given in Figure 6. These oscillograms show that 
even when no crossing of rays is predicted, multiple 
peaks and similar irregularities can still occur, al¬ 
though these features are less pronounced than in 
Figure 10. As was cautioned in Section 9.1, instru¬ 


mental sources for such irregularities must always 
be suspected; however, it seems likely that many of 
the unexplained irregularities found in UCDWR ex¬ 
periments are due in some way to oceanographic 
conditions. 

9 . 2.4 Results of Fourier Analysis 

So far we have discussed the propagation of ex¬ 
plosive sound with little mention of its relation to 
sinusoidal sound waves. There is an important rela¬ 
tion between the two, however. Any pulse of arbi¬ 
trary shape can be approximated as accurately as 
desired by a linear superposition of a sufficiently large 
number of sine waves, of suitably chosen frequencies, 
amplitudes and phases. If the laws of propagation of 
sound are linear in the amplitude of the disturbance, 
as we believe them to be when the amplitude is suf¬ 
ficiently small, we can predict the changes in the size 
and shape of the pulse as it travels through the water 
from the changes in amplitude and phase which each 
of the sine waves would undergo if it were present by 
itself. Conversely, if the behavior of the pulse were 
accurately known, the attenuation and dispersion of 
all the component sine waves could be calculated. 8 

An analysis of this sort can be extremely useful in 

a That this is indeed a practical possibility has been demon¬ 
strated in some experiments at NRL 17 in which the rela¬ 
tive calibration curve of the two hydrophones was determined 
over the range from 5 to 100 kc by Fourier analysis of their 
responses to detonating caps. The resulting curve was found 
to be in excellent agreement with one determined by standard 
CW methods of calibration. 




































SHORT-RANGE PROPAGATION IN DEEP WATER 


207 



SHOT 26 RANGE 580 YARDS 



SHOT 27 RANGE 710 YARDS 



SHOT 29 RANGE 940 YARDS 


Figure 12. Samples of irregular pressure-time curves. 
Source: No. 8 detonating cap at depth 50 feet and 
ranges indicated. Hydrophone at depth 54 feet. Date: 
Apr. 3, 1942. 

elucidating the physical mechanisms operative in the 
propagation of sound in the sea and in predicting the 
response of resonant or band-pass receiving systems 
to explosive sound. 

The representation of an arbitrary disturbance as a 
superposition of sine waves is described mathemati¬ 
cally by Fourier’s theorem. The most useful form of 
this theorem for our present purpose is the integral 


form, which states that if p(t) is any function of an 
independent variable t such that f \p\ 2 dt con¬ 
verges, then for all values of t except points at which 
p is discontinuous 

p(0 = J“ <t>(f)e 2irift df (1) 

where </>(/) = J p{t)e~ 2nf 'dt. (2) 

It can also be shown that 

= (3) 

When these mathematical theorems are applied to 
the pressure p(t) in a pulse of sound, the physical 
interpretation of the results is simple. The integral 
on the left of equation (3), when divided by pc, 
represents the total energy in the pulse per unit area. 
The quantity / represents frequency, measured for 
example in cycles per second, and so the integrand 
\<j>\ 2 on the right of equation (3), when divided by pc, 
represents the energy per unit area per unit frequency 
range. The spectrum level of the pulse at frequency /, 
as measured for example by the energy received from 
it by a receiving system sensitive only to a narrow 
band of frequencies in the neighborhood of /, is 

U(f) = 10 log 10 \<t>(f)\ 2 (4) 

and V will be in decibels per cycle above 1 dyne per 
sq cm, if p was measured in dynes per sq cm and / 
in cycles per second. 

In evaluating the expression (2) for an experi¬ 
mentally obtained pressure-time curve it is of course 
not possible to extend the upper limit of integration 
to infinity; in UCDWR work described in reference 9, 
for example, the oscillographic record obtained only 
lasted for a few hundred microseconds, and over the 
latter part of this range it was hard to estimate the 
position of the zero line accurately. The integrals 
which were actually evaluated were therefore 

f p{t) sin 2irftdt 

Jo 

and f p(t) cos 2-irftdt, 

Jo 

where the origin of time is taken as the time of arrival 
of the first perceptible pressure and where t\ is a few 
hundred microseconds, i.e., is of the order of the 
duration of the traces which were shown in Figures 
7, 10, and 12. Such a curtailment of the upper limit 



208 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



Figure 13. Influence of time 


-. 

\ 

V- 




\V 

I 



\ 

\z 

\ 

\ 



\ 4\; 

\ 

r—s \ 

v 



\ 

\\ 




\ 

\ 


30 1 ---- 

I 5 10 50 100 

FREQUENCY IN KILOCYCLES 


of rise on spectrum of a single pressure pulse. 


of integration results, unfortunately, in omission of 
the surface-reflected pulse when its time of arrival 
exceeds t\. As a result the computed <£ will rise to a 
maximum value as the frequency approaches zero, 
whereas if the surface reflection were included cj> would 
approach a small value or zero. Even if there were no 
surface reflection we should expect the computed 
value of </> to be considerably in error at very low 
frequencies through neglect of the tail of the shock 
wave, which has been discussed in Section 8.5. At 
frequencies large compared with 1 h the neglect of 
the surface reflection is unimportant, although it may 
result in the absence of some small-scale ripples from 
the curve of spectrum level against frequency, which 
would be present if the surface reflection were in¬ 
cluded. When the surface reflection arrives well within 
the time U, of course, the error due to cutting off the 
integration at this time is usually negligible, although 
of course if there are bottom reflections arriving after 
time ti the computed spectral distribution will be 
that which would apply in the absence of a bottom. 

At very high frequencies the spectrum level de¬ 
pends primarily upon the time of rise, and the com¬ 
puted value may be in error if the response time of 
the hydrophone and recording system does not per¬ 
mit faithful reproduction of the rising portion of the 
pulse. The extent to which the time of rise affects the 
spectral distribution is shown by some sample calcu¬ 
lations given in reference 9, for hypothetical pulses, 
which are presented in Figure 13. Note that pulse I, 
which has a vertical rise, lias the highest spectrum 


level at high frequencies. It has been shown by 
mathematicians that the Fourier transform of a finite 
but discontinuous function p is of order 1 / at large 
values of the frequency /, and we should therefore 
expect the spectrum level U for pulse I to decrease 
at 6 db per octave at high frequencies. Similarly, it 
can be shown that a pulse which is continuous but 
has discontinuities in slope, such as IV in the figure, 
should have a spectrum level which decreases at 
12 db per octave at sufficiently high frequencies. The 
spectrum of a perfectly smooth pulse, such as III, 
should decrease still more rapidly. 

Since actual oscillograms usually show irregularities 
in the tails of the pressure pulses, and since these ir¬ 
regularities are usually not very reproducible from 
shot to shot, it is pertinent to inquire how much in¬ 
fluence they have on the spectral distribution. Sample 
calculations for hypothetical pulses have shown that 
the principal effect of a fairly smooth “satellite” peak 
is to introduce irregularities into the curve of spec¬ 
trum level against frequency, without much altera¬ 
tion of its general trend. 9 

Figure 14 gives curves of spectrum level U against 
frequency /, computed from the oscillograms of 
Figure 7. In these curves the irregularities which ap¬ 
pear in the curves directly computed from (4) and (2) 
have been arbitrarily smoothed out; from what has 
been said in the last paragraph these irregularities 
probably have little significance, and eliminating 
them makes it easier to follow the changes in the 
spectrum as the range of the shot is increased. All the 


















SHORT-RANGE PROPAGATION IN DEEP WATER 


209 



Figure 14. Spectral distribution of energy in explosive pulses received at various ranges. The curves shown were 
obtained by smoothing the values calculated by Fourier analysis from a number of shots made on Apr. 3, 1942, at a 
depth of 100 ft and received by a hydrophone at 54 ft. 


curves except the first three show a maximum, since 
in the direct zone the surface reflection makes the 
spectrum level decrease at low frequencies, and in the 
shadow zone a similar effect is produced by the 
oscillatory nature of the pulse, although a separation 
into direct and reflected parts is no longer possible. 
For the first three pulses the integration was not ex¬ 
tended over a sufficiently long time to include the 
surface reflection; if it had been, the curves would 
form a continuous family. The slopes of the curves 
for the pulses received in the direct zone are about 
12 or 13 db per octave at 50 kc; in the shadow zone 
this slope increases to 17 or 18. This change is ol 
course due to the increase in time of rise on entering 
the shadow zone. 

As the range increases in the shadow zone there 
seems to be a slight decrease in the frequency at which 
U is a maximum. This shift, although hardly greater 
than the experimental error, is probably also due to 
the increase in time of rise. If the curves for the direct 
zone had been computed with inclusion of the surface 
reflection, however, they would have shown a trend 


in the opposite direction, since the frequency below 
which the cancellation effect is felt is one for which a 
quarter of a cycle is of the same order as the time 
interval between the direct and reflected pulses, and 
this interval decreases with increasing range. 

The frequency at which the spectrum level for 
shots in the shadow zone is a maximum shows varia¬ 
tions from one day to another which are much greater 
than the variations from shot to shot on a given day. 
Table 3 gives values, taken from reference 9, of this 
frequency observed on a number of days for shots 
well inside the shadow zone. 


Table 3. Frequency of maximum spectrum level in 
the shadow zone. 


Date, 1942 

Frequency of maximum, kc 

March 12 

7.1 

March 19 

8.0 

April 3 

3.3 

April 8 

9.0 

April 9 

15.0 


An interesting application of this type of analysis 
is to compute the attenuation which sound of dif- 


























210 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 




• • • 





• 

• 







•x. • 


40 KC 






500 1000 1500 2000 2500 3000 

RANGE IN YARDS 


Figure 15. Shadow zone attenuation of various fre¬ 
quencies, as obtained by Fourier analysis of explosive 
pulses. Ordinate of each plot is U(f) + 20 log r + 
a(/)r/l,000 referred to an arbitrary level, where U(f) is 
the spectrum level, r is the range in yards, and a(/) 
is the assumed absorption at the frequency /, as given 
in Section 5.22. Hydrophone depth, 54 ft; cap depth 
100 ft; sound conditions as shown in Figure 6. 


ferent frequencies suffers as the range is increased. 
The decrease of spectrum level with increasing range 
is due to geometrical divergence, which can be ap¬ 
proximately but not accurately described by the 
inverse square law, to absorption and scattering, 
which are known to increase with frequency, and to 
the effect of the shadow boundary. To show the latter 
effect more clearly it is convenient to plot the 
quantity 


U (/) + 20 log r + 


a(f)r 

1,000 


against the range r, using values of the attenuation 
constant a(f) appropriate to sinusoidal sound of 
frequency /. This is done for several frequencies in 
Figure 15 for the shots at 100-ft depth of the same 
series as has already been discussed in connection 
with Figures 5, 6, 7, 8, 9, and 14. The plots shown 
have been obtained by applying a correction to the 
points given in reference 9 to bring the assumed at¬ 


tenuation a(f) into line with the more up-to-date 
values given in Section 5.2.2. 

Within the direct zone, the points for all frequencies 
lie roughly along horizontal lines, indicating that 
divergence and normal attenuation suffice at least 
approximately to explain the changes in size and 
shape which the pulse undergoes. Because of the 
deviations from the inverse square law discussed in 
Section 9.2.3 and shown in Figure 5, of course it is 
not to be expected that the points in the direct zone 
will follow a horizontal line very precisely. At the 
shadow boundary the spectrum levels start to de¬ 
crease sharply; it is worth noting that the onset of 
the sharp decrease occurs at 1,400 to 1,000 yd for all 
cases and that, as one would expect, this range agrees 
much better with the value of distance to the shadow 
boundary derived from the time of rise and peak 
pressure in Table 2 than with the value computed 
from the bathythermograph data. Table 4 gives the 


Table 4. Shadow zone attenuation at various fre¬ 
quencies on April 3, 1942. 


Frequency in kc 

Attenuation in db per kiloyard 

1 

21+2 

3 

8 ± 1 

20 

29 ± 2 

40 

27 ± 2 


magnitude of the additional attenuation beyond the 
shadow boundary, as obtained from the slopes of the 
lines in the figure; the probable errors quoted are 
based merely on the internal consistency of the data 
shown in Figure 15, and may not represent the overall 
accuracy of the calculation. The minimum of at¬ 
tenuation at 3 kc is striking, and corresponds of 
course to the maximum shown by the curves of 
Figure 14. As can be seen from Table 3 observations 
on other days, if treated in the same way, would have 
given quite a different dependence of attenuation on 
frequency. It is not known to what extent these dif¬ 
ferences can be correlated with thermal conditions, 
range, and the depths of source and receiver. 

In making a comparison between the attenuation 
suffered by sinusoidal sound and the attenuation of 
the various frequencies making up an explosive pulse, 
we must keep in mind the limitations imposed by the 
short duration of the explosive pulse, or rather by 
the short period of time which is covered by the 
record of it. For example, the measured values of at¬ 
tenuation for a long pulse of sinusoidal sound can be 






































LONG-RANGE SOUND CHANNEL PROPAGATION 


211 


greatly influenced by scattering, whereas sound scat¬ 
tered through any sizable angle would arrive too late 
to be recorded on oscillograms like those of Figure 7. 

9.2.5 Variations 

Ideally it should be possible to determine the magni¬ 
tude and time scale of the fluctuations and variations 
in transmission by firing a number of caps in rapid 
succession from the same place. Unfortunately, no 
systematic experiments of this sort have been carried 
out. In the UCDWR work, 89 a few repeat shots 
were made at intervals of a few minutes; however, the 
number of such repeat shots was curtailed by the 
need for obtaining data at different ranges and depths 
in a time short enough so that oceanographic condi¬ 
tions could be assumed constant. Most of the ma¬ 
terial in the following paragraphs represents infer¬ 
ences obtained when some of the oscillograms for 
these experiments were restudied in the course of 
preparing material for the present report; because of 
the paucity of the data, these inferences must be re¬ 
garded with caution. 

In isothermal water, peak pressures from succes¬ 
sive shots seem to vary but little out to ranges of 
over 1,000 yd. Most of the shots studied were con¬ 
sistent to within 1 or 2 per cent, though occa¬ 
sional shots deviated by 5 or 10 per cent. These 
variations are of the same order as those which are 
found at short ranges and attributed to nonuni¬ 
formity of the caps themselves. The fact that they 
are so small is evidence of the uniformity of adjust¬ 
ment of the hydrophone and recording system. 

When the cap is in the thermocline and the hydro¬ 
phone in an isothermal layer above it, or when there 
is a negative temperature gradient at all depths, the 
fluctuations in peak pressure seem to be distinctly 
greater; for these cases successive shots at a few 
hundred yards range often differ by 20 per cent or 
more. In the experiments with single-frequency 24-kc 
sound, which were reported in Section 7.1.1, the 
fluctuation was found to decrease somewhat if the 
hydrophone was placed beneath the thermocline. 
Apart from the fact that neither the evidence on 
explosive sound nor that on single-frequency sound 
is based on an adequate number of samples, the ap¬ 
parent contradiction may be readily explained by the 
fact that in 24-kc single-frequency work the surface- 
reflected signal usually cannot be resolved from the 
direct signal, while in the experiments reported here, 
these two signals are generally received one after the 


other. Both in the present case and in the preceding 
the surface-reflected pulse seems to be a little more 
variable than the direct pulse, although the evidence 
for this is not very conclusive because of the irregu¬ 
larities, real and instrumental, which are present in 
the tail of the direct pulse on which the reflection is 
superposed. 

Beyond a shadow boundary successive shots are 
often surprisingly consistent. The difference between 
the first pressure peak and the first trough, for ex¬ 
ample, has been observed in several cases of repeat 
shots to be reproducible to within 20 per cent or so 
although at least one case of a much larger fluctua¬ 
tion has been observed. 

Figure 16 shows some typical oscillograms of shots 
made a few minutes apart, for three types of trans¬ 
mission conditions. One must be cautious in attrib¬ 
uting physical reality to all the differences in detail 
which appear in successive oscillograms of this sort. 
For example, it has been demonstrated that slight 
changes in the orientation of a hydrophone from one 
shot to the next can sometimes produce considerable 
changes in the recorded pressure-time curve. 1 Other 
variable factors mentioned in reference 1 which can 
have an appreciable effect include scattering of sound 
by supports and other bodies near the hydrophone, 
and the possible presence of bubbles or other foreign 
matter on the hydrophone. Thus differences between 
successive records may or may not be real. On the 
other hand, any feature which is consistently repro¬ 
duced in all records made under a given set of condi¬ 
tions, and for which an instrumental origin can be 
ruled out by virtue of its nonappearance under most 
other conditions, is probably a reproducible char¬ 
acteristic of the true pressure-time curves under the 
given conditions. A feature of this type is the shape 
of the first cycle or so of the oscillatory pressure-time 
curve in a shadow zone, as shown in Figure 16C. 

9.3 LONG-RANGE SOUND CHANNEL 
PROPAGATION IN DEEP WATER 

It has long been known that explosive sound from 
a charge of moderate size can be detected at ranges 
of the order of tens of miles, and this fact has re¬ 
ceived practical application in acoustic position find¬ 
ing. 18 ’ 19 As will be shown later, ranges of thousands 
of miles can be achieved by proper arrangement of 
source and receiver. It is to be expected that many 
of the characteristics of the signals received at long 
ranges will be determined by refraction and by re- 



212 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



A SHOTS 9 AND 10, FEB 26, 1942 

DEPTH OF CAP 80 FEET, DEPTH OF HYDROPHONE II FEET, RANGE 1200 YARDS 


TRANSMISSION OF ISOTHERMAL WATER 



B SHOTS 2 AND 4, APRIL 3, 1942 

DEPTH OF CAP 100 FEET, DEPTH OF HYDROPHONE 54 FEET, RANGE 610 YARDS 
DOWNWARD REFRACTION TRANSMISSION IN DIRECT ZONE 


C SHOTS 25 AND 26, MARCH 19, 1942 

DEPTH OF CAP 50 FEET, DEPTH OF HYDROPHONE 10 FEET, RANGE 1520 YARDS 
DOWNWARD REFRACTION, TRANSMISSION INTO SHADOW ZONE 


Figure 16. Typical examples of the reproducibility of records of explosive pulses for shots fired a few minutes apart. 




213 


LONG-RANGE SOUND CHANNEL PROPAGATION 


tj 


a. 

UJ 

o 


5000 


10,000 


15.000 



4850 4950 505 0 0 2 0 4 0 60 80 

VELOCITY OF S0UN0 IN FT PER SEC RANGE IN KILOYARDS 


Figure 17. Velocity distribution and schematic ray paths for a typical deep sound channel. 


flection from the bottom. Leaving the detailed con¬ 
sideration of bottom reflections until Sections 9.4.1 
and 9.4.2, we shall consider here a number of phe¬ 
nomena which are due to the presence of a “sound 
channel” at a certain depth in the ocean. A sound 
channel can be briefly described as a stratum of the 
ocean within which the velocity of sound first de¬ 
creases with increasing depth, passes through a min¬ 
imum, and then increases. The importance of such a 
region is due to the fact that if a source of sound is 
placed in it, all rays emitted within a certain range of 
initial directions will remain confined to the sound 
channel, executing periodic oscillations in depth. 

All, or nearly all, of the experimental results which 
have been obtained so far on long-range transmission 
in deep water can be interpreted in terms of ray 
paths. Ray paths which lie in a sound channel are of 
especial importance, and since these rays have rather 
complicated characteristics, Section 9.3.1 will be de¬ 
voted to a theoretical discussion of them. By using 
the facts established there as a foundation, the ex¬ 
perimental results to be given in Section 9.3.2 can be 
concisely discussed and interpreted. 

9.3.1 Deep Sound Channels 

Sound channels may occasionally occur at shallow 
depth, when there is either a layer of isothermal water 
or a layer with a positive temperature gradient under¬ 
neath a surface layer in which the temperature 


gradient is negative. The more common configura¬ 
tion of an isothermal layer immediately beneath the 
surface is, moreover, very similar to a sound channel, 
in that many rays undergo alternate upward refrac¬ 
tion and surface reflection, and so remain confined to 
the isothermal layer out to indefinitely large ranges. 
Of more importance, however, for long-range trans¬ 
mission in deep water, is a deep sound channel which 
always occurs, except in polar regions, at a depth of 
the order of 4,000 ft or less. This deep sound channel 
is due to the fact that at all ordinary latitudes there is 
an extensive thermocline within which the tempera¬ 
ture gradient is negative and below which the tem¬ 
perature gradient is so slight that the effect of in¬ 
creasing pressure suffices to make the velocity of 
sound increase with depth. 

Figure 17 shows velocity minima of both shallow 
and deep types, and several ray paths emanating from 
a source located at the depth of the lower minimum. 
It will be convenient to refer to the ray which be¬ 
comes horizontal at the depth of minimum velocity 
as the “axis” of a sound channel. If the thermal 
gradient is discontinuous, as in Figure 17, the num¬ 
ber of different rays connecting any two points on the 
axis is infinite, since the range per cycle, that is, the 
horizontal distance traversed by a ray while travers¬ 
ing 1 c of its oscillation in depth, approaches zero 
as the ray approaches the horizontal. This is illus¬ 
trated by the three rays labeled I. If the velocity- 
depth curve is smooth near the minimum, this will 





























214 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



ANGLE AT WHICH RAY CROSSES AXIS 
OF SOUND CHANNEL, IN DEGREES 


TYPE I-UNREFLECTED RAYS 

TYPE H-SURFACE REFLECTION AND UPWARD REFRACTION 
TYPE m-BOTTOM-REFLECTED RAYS 

- HORIZONTAL RANGE TRAVERSED BY RAY IN EXECUTING 

ONE COMPLETE CYCLE OF ITS VERTICAL OSCILLATION 

-MEAN HORIZONTAL VELOCITY EQUAL TO ABOVE RANGE 

DIVIDED BY TIME REQUIRED FOR ONE CYCLE 

Figure 18. Mean horizontal velocity and range per 

cycle for rays oscillating about a sound channel. 

not be true. In any case, however, whether source 
and receiver are on the axis of the sound channel or 
not, the number of different rays connecting them 
increases with increasing horizontal range. The num¬ 
ber of such rays decreases, however, with increasing 
distance of source or receiver from the axis. If either 
source or receiver is too far from the axis of the sound 
channel, no ray can get from source to receiver with¬ 
out reflection from the surface or the bottom. 

To study these phenomena quantitatively, and to 
compute times of arrival for the various rays, curves 
like those shown in Figure 18 are very helpful. What¬ 
ever its point of origin, any ray which traverses the 


sound channel can be characterized by the angle at 
which it crosses the axis of the sound channel; any 
two rays which cross the axis at the same angle must 
be congruent, differing only by a horizontal displace¬ 
ment. Figure 18 shows how the horizontal range per 
cycle and the mean horizontal velocity, that is, the 
quotient of horizontal range per cycle by time per 
cycle, depend upon the angle of crossing the axis. 

For angles less than a certain critical value, equal 
to 12.2 degrees in the example shown, the ray 
oscillates up and down without reaching either the 
surface or the bottom. For rays of this type (type I 
in Figure 17) it will be seen that the mean horizontal 
velocity is least for the axial ray and increases as the 
angle with the horizontal, and hence the range per 
cycle, increases. The consequences of this are espe¬ 
cially interesting when both source and receiver are 
on the axis of the sound channel. For this case the 
first impulse to arrive will come along a ray for which 
the number of oscillations in depth has the smallest 
value consistent with avoidance of surface and bot¬ 
tom reflections. When the range is small, this ray 
will have only one half-cycle between source and re¬ 
ceiver, but with increasing range more and more half¬ 
cycles are required, since the range per complete cycle 
can never be greater than a certain value, equal to 
85 kyd in Figure 18. Rays with more and more 
oscillations will arrive later and later, and if for the 
moment we ignore reflected rays, the last one to 
arrive will be the straight axial ray. Thus, the early 
arrivals will be separated by considerable intervals 
of time, but later arrivals will be closer and closer to¬ 
gether, finally merging into an unresolvable cre¬ 
scendo, followed, if we neglect reflected rays, by a 
sudden silence. Figure 19A shows the times of arrival 
of these sound channel rays, as computed from the 
data in Figure 18 for a particular value of the range. 

The total time between the first and last of these 
arrivals can be computed from the spread in mean 
horizontal velocities for the sound channel rays; for 
the case plotted in Figure 18 the total time in seconds 
comes out to be 0.012 times the range in miles. 

It will be noticed that, the early arrivals in Figure 
19A come in groups of three. The explanation of this 
is shown schematically in Figure 20 for the simplest 
case of the first arrivals at a very short range. Each 
oscillating ray travels much farther in a lower half- 
cycle than in an upper one; consequently the mean 
horizontal velocity of a ray between source and re¬ 
ceiver will be principally a function of the amplitude 
of its lower half-cycles which in turn is principally 









215 


LONG-RANGE SOUND CHANNEL PROPAGATION 


time in seconos 



A 



Figure 19. Times of arrival for the various rays connecting two points on the axis of a sound channel. Range, 400 kyd = 
197 miles. Velocity-depth curve assumed same as for Figures 17 and 18. The numeral below each arrival gives the num¬ 
ber of lower half-cycles in the corresponding ray path. The zero of time is taken as the time of arrival of the axial ray. 


dependent on the number of lower half-cycles which 
occur during the passage from source to receiver. In 
the example of Figure 20, there are four rays which 
have two lower half-cycles between source and re¬ 
ceiver; however, two of these four, namely, the ones 
with two upper half-cycles, arrive at the same time so 
there will be only three resolvable arrivals. When 
source and receiver are at different depths, the rays 
analogous to those in Figure 20 will all arrive at 
different times, and the hydrophone will receive 
pulses in groups of four. For the later arrivals, the 
upper half-cycles have more nearly the same travel 
time as the lower half-cycles, and the pulses no longer 
arrive in clearly separated groups of three. 


When either the source or the receiver is at some 
distance from the axis of the sound channel, the 
piling-up effect shown in Figure 19A will not occur, 
since only a limited number of rays will be possible 
between source and receiver. 

So far we have not considered sound which arrives 
at the receiver by paths involving surface or bottom 
reflection. The rays which undergo surface reflection 
and upward refraction without reaching the bottom 
(type II in Figure 17) have mean horizontal velocities 
which, according to Figure 18, are slower than the 
fastest unreflected, or type I rays, but faster than the 
axial ray. Thus the arrivals for rays of this sort are 
mixed in with those of type I, but when source and 


































216 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


RANGE 



RANGE 



COINCIDENT IF, AS SHOWN HERE, SOURCE 
AND RECEIVER ARE AT THE SAME DEPTH 


RANGE 




Figure 20. Grouping of arrival times for sound chan¬ 
nel rays. The four ray paths shown each have two 
lower half-cycles, and the corresponding times of travel 
are therefore close together, so that the four arrivals 
form a group. 

receiver are both on the axis they cease before the 
“piling up” of the sound channel rays. This is shown 
in Figure 19B for the particular set of conditions 
chosen for that figure. 

The bottom-reflected rays (type III in Figure 17) 
have times of arrival which are also interspersed 
among those of type I, but which continue after the 


latter have ceased. Figure 19C shows these arrivals 
for the example treated. The grouping for these rays 
is again in threes or fours, according to whether 
source and receiver are at the same or different 
depths. 

Let us now consider the energies and intensities of 
the system of impulses arriving at the receiver. First 
of all, it may be noted that all the energy emitted 
by the source in directions giving rays of type I, i.e., 
for the cases of Figure 18 in directions within + 12.2° 
of the horizontal, is propagated along the sound 
channel and cannot disappear except by volume 
absorption or scattering in the water. If the latter 
processes are neglected, the total energy in the 
system of impulses transmitted by these unreflected 
rays, that is, the system exemplified by Figure 19A, 
must be inversely proportional to the horizontal 
range. This contrasts with propagation in an infinite 
homogeneous medium, where the energy given to a 
receiver varies inversely as the square of the distance. 
The intensities of the individual arrivals can be calcu¬ 
lated in the usual way from ray theory, which should 
be applicable to the earlier arrivals, before there is 
appreciable overlapping of consecutive pulses. It will 
be apparent after a little pondering on Figure 17 that 
in general these individual intensities must vary ap¬ 
proximately as the inverse square of the range, the 
slower rate of decay of the total energy being due 
to the increase in the number of arrivals as the range 
increases. For certain positions of source and re¬ 
ceiver, however, some of the arrivals may have an 
anomalously high intensity due to the fact that two 
rays of infinitesimally different initial inclinations 
are tangent at the receiver. This condition will be 
more closely approached for the latest soimd channel 
arrivals to reach the receiver than for the earlier ones, 
and accordingly these latest arrivals should be the 
strongest. 

The energy traveling along paths involving reflec¬ 
tion from the surface or the bottom is channeled in a 
similar manner. The bottom-reflected rays, however, 
lose a considerable part of their energy at each re¬ 
flection, and therefore die out more rapidly with in¬ 
creasing range than the others. (See Figure 21 of 
Section 9.3.2 and Section 9.4.1.) For the same reason 
successive arrivals of this type have progressive^ de¬ 
creasing intensities. 

9 . 3.2 Experimental Results 

Two series of experiments on long-range transmis¬ 
sion in deep water have been conducted by WHOI. 20 ’ 21 









LONG-RANGE SOUND CHANNEL PROPAGATION 


217 


1 RADIO SIGNAL 

2 SHALLOW HYDROPHONE 

3 DEEP HYDROPHONE 


1 / 




ii ** 

St- 

I, ',n , 

-- 

-- ... ... s=t - r J- 

***** ... . *~ . .. .... 

->■> . 1.^ 

T « 

-Si LU 

2 






H 




- Ml 

r*—‘ 

- 


_ 





3 


*14tr 







*asi > K.uf 

// / ' #>/h * 

T * m T 4 4 — - * 

<««•) ti:** 


•/« 

irm 

Ti '■ r* t r • * T* - 






2 3 4 *V 

IOIO IOIO ^—END OF SOUND CHANNEL ARRIVALS 

FREQUENCY A SHOT 34, RANGE 50 MILES 

IN C 



IOIO IOIO END OF SOUND CHANNEL ARRIVALS-^ 

FREQUENCY B SHOT 43, RANGE 300 MILES 


INC 


Figure 21. Typical records of explosive sound received at long ranges. Times marked along top of each oscillogram 
are in seconds. Curves at left give relative amplitude response of each channel to the various frequencies. 


The first series was conducted just outside the Ba¬ 
hama Islands and consisted in the recording of im¬ 
pulses from shallow explosions on two hydrophones, 
one at shallow depth and the other at 1,600 ft, at 
various ranges all less than 30 miles. The second 
series was made some time later in regions extending 
northeastward and eastward from the same locality. 
In addition to the shallow shots, explosions at 4,000 ft, 
near the axis of the main sound channel, and a hydro¬ 
phone near this depth as well as a shallow one were 
used. The ranges for this series extended out to 900 
miles. b Data pertaining to the conditions of these ex¬ 
periments are given in Table 5. The velocity-depth 
curve for the first series is the one shown previously 
in Figure 17; that for the second series is very 
similar, and the curves of Figure 18 can be applied 
with little error to either series. 

Figure 21 shows some typical records obtained in 
these experiments along with thumbnail sketches of 
the frecpiency response characteristics of the various 
recording channels used. The following paragraphs 
point out a number of features of these records which 
agree with the predictions of ray theory as outlined 
in Section 9.3.1. 

b More recent experiments which have not yet been re¬ 
ported in full have yielded detectable signals at a range of 
2,300 miles. 


Table 5. Experimental arrangements used in long- 
range transmission studies by WHOI. 



First series 

Second series 


(from refer- 

(from refer- 


ence 21) 

ence 20) 

Depth shallow hydrophone in 



feet 

80 

80 

Depth deep hydrophone in feet 

1,600 

3,500 

Charge weights and depths 

lb, 50 ft 

H lb, 50 ft 

4 lb, 4,000 ft 
200 lb, 300 ft 

Ranges, nautical miles 

2.7-26 

20-900 

Depths of sound channels in feet 

75, 4,500 

4,100 (average) 

Depths of water in feet 

16,000 

15,000-18,000 


(usually) 


Identification of the paths by which the various 
pulses arrive is usually difficult because of the large 
number of arrivals and because the predicted time 
for any arrival can be appreciably influenced by un¬ 
certainties in the depths of source and receiver and 
by small variations of the velocity-depth curve along 
the route. However, the general appearance of the 
records is very much as one would expect from the 
considerations given in Section 9.3.1. Thus, in 
Figure 21B the arrivals at the deep hydrophone come 
in groups of four while for the shallow hydrophone 
the four pulses show up as two, each of which is pre¬ 
sumably double but unresolved because of the short- 
























































218 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


ness of the interval between any pulse and its reflec¬ 
tion from the surface near the hydrophone. In 
Figure 21A a similar grouping occurs for the bottom- 
reflected pulses, which continue after the last sound 
channel arrival, but the range for this case is so short 
that the sound channel arrivals have not yet had time 
to form into well-defined groups. 

In the records obtained when both source and re¬ 
ceiver were shallow, each theoretical group of four is 
of course entirely unresolved and appears as a single 
pulse. 

When source and hydrophone are both deep the 
total duration of the sound channel arrivals, that is, 
the time interval between the first arrival and the 
last arrival via the sound channel is found to agree 
nicely with the duration predicted by Figure 18. At 
the longer ranges the last sound channel arrival is 
easily spotted by the conspicuous “piling-up” effect 
which occurs just before it, as shown in Figure 2IB. 
At shorter ranges, however, as in Figure 21A, the 
last sound channel arrival can only be identified by 
its high intensity and the fact that, like all the ar¬ 
rivals which do not involve reflection from the sur¬ 
face, it is absent from the record of the shallow hydro¬ 
phone. For short ranges the number of sound channel 
arrivals is small because of the short range and the 
fact that neither source nor receiver is on the axis 
of the sound channel, and the bottom reflections, 
which come in both before and after the last sound 
channel arrival, may mask the abrupt termination of 
the sound channel arrivals even when a piling up 
occurs. At ranges beyond about 300 miles, on the 
other hand, the bottom reflections become so weak 
that they no longer appear on the records, and the 
piling up of the sound channel rays is followed by 
sudden silence. This disappearance of the reflected 
pulses is of course due to the fact that according to 
Figure 18 any ray traveling by bottom reflections 
cannot go more than about 70 kyd between successive 
reflections; and since an appreciable amount of 
energy is lost at each reflection, pulses traveling 
along such rays are much more rapidly attenuated 
than those which travel in the water alone. 

According to Table 5 at the time of the first series 
of experiments there was a shallow sound channel 
with its axis at a depth of about 75 ft. That trans¬ 
mission to considerable distances near the surface 
was possible at this time is shown by a comparison 
of the velocity of propagation of the first arrival with 
the velocity for the bottom-reflected rays. The ratio 
of these velocities is found to agree nicely with the 


ratio of the velocity of sound at the surface to the 
velocity given by Figure 18 for the particular bottom 
reflection studied, showing that the first arrival does 
indeed come by a path lying entirely in the region 
near the surface. 

The most significant results obtained in these ex¬ 
periments have to do with attenuation and with the 
reflection coefficients of bottom and surface. To study 
quantitatively the variation of intensity with dis¬ 
tance and with number of bottom reflections, some 
sort of measurements must be carried out on the os¬ 
cillograms. The most obvious thing to measure would 
be the peak pressures or moment urns of individual 
arrivals. However, in many cases the individual ar¬ 
rivals of a group were not completely resolved, and in 
all cases the pressure-time curves may have been 
distorted by small-angle scattering or off-specular re¬ 
flection. For these reasons it was concluded that the 
most suitable characteristic of the records from which 
to estimate attenuations and reflection coefficients is 
the energy of a poke, or of a group of two pokes, 
rather than the peak deflection, this energy being as¬ 
sumed to be a constant times the integral of the 
square of the deflection. One may hope that this 
quantity will represent a suitably weighted average 
of the spectrum level of the pressure pulse in the 
water in the region of frequencies covered by the re¬ 
cording channel being used. It is of course not strictly 
true that the “energy” measured in this way on an 
oscillogram represents this weighted average, since, 
for example, the phase of the transient disturbance 
produced by the first arrival at the time of the second 
arrival will determine whether the second arrival 
increases or decreases the amplitude. However, we 
may expect that, the desired correspondence will be 
valid for an average over many pokes. 

Because of the very large ranges covered by the 
second series of experiments, it was possible to meas¬ 
ure the very small attenuations suffered by sound at 
the comparatively low frequencies to which the re¬ 
ceiving channels responded, frequencies at which no 
other measurements of attenuation have been ob¬ 
tained. The results, based on the total energy of all 
the sound channel arrivals taken together, are sum¬ 
marized in Table 6. The data beyond about 200 miles 
are fairly consistent, as the sample plot given in 
Figure 22 shows. At shorter ranges the measured 
energies vary erratically, perhaps because the number 
of sound channel rays is too small to give a uniform 
spatial distribution of energy. 

The interpretation which should be given to these 



BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


219 


Table 6. Attenuation coefficients for explosive sound received in various frequency bands. 


Location of shots 

Frequency limits of 
recording channel 

6 db below peak 
sensitivity in 
cycles per second 

Attenuation in 
db per kiloyard 

Number of 
observations used 

Spread of 
ranges used 
in kiloyards 

Line from Latitude 26° N, Longitude 76° W 

22-175 

0.005 

5 

400-1,600 

to Latitude 39° N, Longitude 67° W 

2,300-10,000 

0.013 

4 

200-1,000 

Line running east from Latitude 25° N. 

14-75 

0.025 

5 

200-550 

Longitude 76° W 

56-350 

0.043 

4 

300-550 


600-4,000 

0.035 

4 

300-550 


56-350 

0.050 

4 

300-550 


attenuation figures is rather uncertain, since the re¬ 
ceiving channels each cover a fairly broad band and 
since it is likely that the attenuation varies strongly 
with frequency. Moreover, it can hardly be decided 
yet whether the attenuation is due to absorption, to 
scattering, or to variations in the depth of the sound 
channel with geographic position. The latter factor 
would have an influence on the observed intensities 
similar to that of changing the depth of the explo¬ 
sion, the important variable being merely the distance 
of the explosion from the axis of the sound channel. 
In spite of all these uncertainties, however, the figures 
in Table 6 probably do give a significant upper limit 
to the order of magnitude of the absorption at sonic 
frequencies. 

Measurements of a similar sort carried out on the 
bottom-reflected pulses of the first series of experi¬ 
ments give values for the reflection coefficient of the 
bottom which will be presented later in Table 7 (see 
Section 9.4.1). In these experiments no difference 
could be noticed between pulses of the same group 
whose ray paths differed by one in the number of sur¬ 
face reflections undergone. This shows that the reflec¬ 
tion coefficient of the surface was unity, to within an 
accuracy of five or ten per cent, at the angles of inci¬ 
dence involved, which ranged from nearly normal 
incidence down to about 10 degrees from the hori¬ 
zontal. 

It has been suggested that triangulation based on 
the times of the last sound channel arrivals at several 
stations might be of practical use in the accurate loca¬ 
tion of a boat or plane on the ocean. Extrapolation of 
the intensities so far measured for the crescendo 
formed by the last sound channel arrivals suggests 
that a few pounds of high explosive may be heard 
above background at ranges of ten or twenty thou¬ 
sand miles or more, if shoals or land masses do not 
intervene to cast a shadow. 20 


RANGE IN KILOYARDS 



TRAVEL TIME OF LAST SOUND CHANNEL ARRIVAL 
IN SECONDS 

Figure 22. Sample plot of the dependence on range of 
the total energy recorded for all sound channel arrivals. 
Source: 4-lb TNT bomb. Location of shots: line from 
latitude 26° N, longitude 76° W, to latitude 39° N, 
longitude 67° W. Recording channel within 6 db of 
peak sensitivity in range 22 to 175. Line shown cor¬ 
responds to attenuation of 0.0050 db/kyd. 

9.4 BOTTOM REFLECTION AND 
SHALLOW-WATER TRANSMISSION 

The bottom of the ocean can influence the trans¬ 
mission of an explosive pulse in several closely related 
ways. When a pulse traveling through the water 
strikes the bottom, it is partly reflected and partly 
transmitted. If the bottom consists of two or more 
successive strata with different acoustic properties, 
the transmitted pulse may itself be partially reflected 










































220 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


and partially transmitted at the boundaries between 
the strata, and a complicated sequence of multiple 
reflections may take place. Finally, the pulse may be 
transmitted horizontally through the bottom, the 
disturbance of the bottom at each point being accom¬ 
panied by a corresponding disturbance in the water. 
In this phenomenon the impact of the explosive wave 
on the bottom below the charge sets the bottom into 
vibration, and this vibration is propagated radially 
outward like a surface wave on the water, or, to use 
a more accurate analogy, like a surface-bound earth¬ 
quake wave, its frequency and velocity being in¬ 
fluenced, however, by the water overlying the 
bottom. 

The three following subsections deal in turn with 
the simpler and the more complex aspects of these 
phenomena. Section 9.4.1 treats ordinary reflections, 
using the concept of sound rays, and discusses arrival 
time data for certain parts of the “earthquake wave,” 
since these data can also be interpreted in terms of 
rays. Sections 9.4.2 and 9.4.3 discuss the detailed 
form of the pulses transmitted by the “earthquake 
wave,” which can be understood only by abandoning 
the ray concept and treating water and bottom as a 
single dynamical system. 

In the theoretical portions of all these sections it 
will be assumed for simplicity that the bottom is 
smooth and horizontally stratified; and an effort will 
be made to interpret the experimental material in 
terms of this idealization. It must be remembered, 
however, that there may often be small-scale ir¬ 
regularities in the bottom which will scatter the ex¬ 
plosive pulse, and large-scale departures from hori¬ 
zontal stratification, which will complicate the trans¬ 
mission phenomena. 

9.1.1 Reflection Coefficients and 
Times of Arrival 

When a pulse of sound strikes a plane boundary 
between two media of different acoustic properties, 
the reflected pulse has a lower amplitude than the 
incident pulse and in general a different phase. A 
theoretical derivation of the amplitude and phase 
relations to be expected at the boundary between 
two ideal fluid media has been given in Section 2.6.2. 
Actual ocean bottoms may differ in their properties 
from the ideal media considered there, however. To 
describe completely the reflecting properties of a 
given bottom, one should specify the amplitude re¬ 
duction and phase shift for all frequencies and all 


angles of incidence. An equivalent description, which 
could be related to this by the methods of Fourier 
analysis (see Section 9.2.4) would be provided by 
recording the exact form of the reflected pressure¬ 
time curve for an explosive pulse for all angles of inci¬ 
dence. So far, however, no pressure-time curves have 
been recorded for explosive pulses reflected from the 
bottom. The only quantitative data on bottom re¬ 
flections which are available are those obtained at 
WHOI' 7 in connection with the long-range propaga¬ 
tion studies discussed in Section 9.3.2. These data 
will now be described. 

As mentioned in Section 9.3.2, the series of experi¬ 
ments for which analyses of bottom reflections were 
carried out was made using a shallow hydrophone at 
80 ft and a deep hydrophone at 1,600 ft, with shots 
of 3^-lb TNT fired at depths of the order of 50 ft at 
ranges up to 30 miles. Two recording channels with 
different frequency responses were used for the shal¬ 
low hydrophone, and five for the deep hydrophone. 
The reflection coefficients of the bottom were deter¬ 
mined for each of these channels by making plots 
similar to that in Figure 22 for the pulses undergoing 
respectively one, two, and three bottom reflections 
and then measuring the vertical displacements be¬ 
tween the lines corresponding to different numbers of 
reflections. The values obtained are given in Table 7. 


Table 7. Reflection coefficients for the bottom in the 
region near Latitude 26°46' N, Longitude 76°25' W. 


Hydrophone 

Recording 

channel 

Frequency limits 
of recording channel 

6 db below peak response 

Average 

reflection 

coefficient 

80 ft 

2 

1,900-6,200 

0.04 


3 

240-2,400 

0.36 

1,600 ft 

4 

42-230 

0.72 


5 

42-230 

0.60 


6 

160-2,400 

0.33 


7 

200-2,800 

0.33 


This method of analysis, while probably the best that 
can be applied to the data available, is rather crude 
in that the angle of incidence of the rays on the 
bottom changes with range and also with the order of 
the reflection; if, as is often the case, the reflection 
coefficient varies strongly with angle of incidence, 
only a vague average over a range of angles will be 
obtained. 

The values given in Table 7 suggest a decrease of 
reflection coefficient with increasing frequency, an 
effect which would not occur at a plane boundary 
between two ideal acoustic media. Unfortunately 







BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


221 


these results cannot be compared with data for 
sinusoidal sound, since the character of the bottom 
in the locality of the experiments is at present un¬ 
known, and since measurements with sinusoidal 
sound at low frequencies are not very complete. 

Information can also be obtained from explosive 
sound regarding the geological strata beneath the 
bottom. Figure 23 shows a typical ray diagram for 



Figure 23. Ray paths in a stratified bottom. 

sound originating in the water over a stratified bot¬ 
tom, in which each successive layer has a higher 
sound velocity than the one above it. Without 
bothering about the detailed form of the pressure 
pulse received at a distant hydrophone, a subject 
which will be discussed fully in the two following sub¬ 
sections, we may study the way in which the time of 
arrival of the first measurable disturbance varies with 
the range from the explosive to the hydrophone. If 
this range, EH in Figure 23, is sufficiently short, the 
first disturbance will arrive by a path which lies en¬ 
tirely in the water. But if EH is greater than a certain 
value r 0 , which depends upon the depths of source and 
hydrophone and upon the velocity of sound in the 
top layer of the bottom, sound traveling along the 
ray EABH will arrive before the direct sound wave 
through the water; in such a case the fact that the 
sound velocity over the path AB is greater than that 
in the water more than compensates for the fact that 
EABH is longer than the direct path EH. To find 
out when this occurs, let c be the velocity of sound in 
the water (assumed uniform for simplicity), Ci the 
velocity in the top layer of the bottom, v the hori¬ 
zontal range, and z the height of explosive and hydro¬ 
phone above the bottom, assuming for simplicity 
that both are at the same level. We shall first show 
that the positions of A and B, which minimize the 
time of travel, are those for which the angles EAB 
and ABH obey the refraction law of ray theory, and 


shall then derive an expression for the value of r at 
which the time of travel via EABH becomes shorter 
than via the direct route EH. 

The time required for a pulse of sound to travel 
a path such as EABH in Figure 23 is 

2 (esc 6 e + CSC 0 h ) r - z(cot 0 e + cot 0 k ) 
t = -■—• -\ -(5) 

C Ci 


If this time has a minimum as the position of point A 
is varied, this minimum must occur when (U/dO e = 0, 
that is, when 

z z 

— esc 0 e cot 6 e -|— esc 2 6 e = 0, 

C Ci 

which is equivalent to 

c 

cos 0 e = —■ 

Cl 

This is the well-known expression for the angle at 
which the transition from refraction to total reflec¬ 
tion occurs. Similarly, the requirement that t be a 
minimum with respect to displacements of point B 
gives 

c 

cos 6), = cos 0 e = — • (()) 

Ci 


Eliminating the angles from equation (5) by use of 
this relation, we have for the time of arrival by the 
shortest path through the bottom 


2z + r 2z c/ci 

cV l - &/c\ ^ CiV 1 - &/c\ 

r 2zVc\ - c 2 

_ _|-- 

Cl CCi 


(7) 


This equals the arrival time r/c of the direct pulse 
through the water when 


2z 


Cl + C 
Cl - c 


( 8 ) 


Now if the time interval between the explosion and 
the first signal at the hydrophone is plotted against 
the range r, the graph will start out as a straight line 
passing through the origin and of slope 1/c; and at 
the range given by equation (8) the slope will change 
abruptly to 1/ci. Thus all the quantities c, Ci, and h 
could be determined from this plot. If the plot is 
continued to larger values of r, another abrupt 
change of slope may occur when the travel time via 
a path EMNQRH lying partly in a denser stratum 
(medium 2) becomes shorter than via EABH. If the 
bottom contains still deeper strata with higher sound 
velocities, further changes of slope will occur. By 
methods similar to those outlined above, the depths 


































222 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


RANGE IN YARDS 


0 800 1600 2400 3200 4000 4800 5600 

l-i- 1 - 1 - 1 - 1 - 1 - 1 - 1 -t- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -r- 1 -' 



Figure 24. Typical plot of travel time against range showing layer structure of the bottom. Location near Solomons, 
Md., at mean depth of 52 ft, ch arge and hydrop hone both resting on bottom. Lines cross at r/c = 1.08 seconds. Depth of 
upper layer in bottom = r/c y/c-i — ci/c; + Ci = 1,240 feet, where r = range at which lines cross, c = velocity of sound 
in water, Ci = velocity of sound in upper layer of bottom, and c 2 = velocity of sound in lower layer of bottom. 


as well as the velocities of all these strata can be de¬ 
termined from this plot of arrival times. This type 
of analysis has long been familiar in geophysical 
prospecting. 

Figure 24 shows a typical plot of arrival times con¬ 
structed from some of the data obtained by WHOI, 22 
with the layer depths and velocities deduced from it. 
The shots were made with both the charge and the 
hydrophone on the bottom, so the depth of the upper 
layer of the bottom can be calculated from equation 
(8) by replacing c by Ci and Ci by c 2 . When more than 
two layers are involved, the plot of times of first ar¬ 
rival will still consist of straight-line segments, but 
the calculation of the depths of the second and 
deeper layers involves more complicated formulas 
in that case. 

The representation of the plotted points by two 
straight lines is fairly easy for this case; how T ever, 
data are often obtained for which the times of first 
arrival seem to form an almost smooth curve. This 
may sometimes be due to the absence of any well- 


defined layer structure in the bottom, as might be 
the case for example for a thick mud bottom whose 
compactness increases continuously with depth. It 
will be shown in Section 9.4.3, how r ever, that there 
are many cases where there are recognizable layers 
in the bottom but where fluctuations of one sort or 
another prevent them from being accurately identi¬ 
fied from mere arrival time data. For such cases the 
proper choice of straight lines to fit a plot such as 
Figure 24 may sometimes be facilitated by a study 
of the predominant frequencies in the first and 
subsequent arrivals (see Figure 32, Section 9.4.3). 

9.4.2 Simplified Theory of Normal 
Modes 

We have seen in the preceding Section 9.4.1 that if 
the range is sufficiently long compared with the dis¬ 
tances of source and receiver above the bottom, the 
first sound to arrive must come by a path lying 
within the material of the bottom over most of the 









































BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


223 


distance, as shown in Figure 23. One might at first 
suppose that refraction of this sort would be similar 
to refraction in the water alone, and that the re¬ 
ceived pulse would be a replica of the pressure wave 
emitted by the source, with an intensity which could 
be calculated by ray theory. It is easily shown, how¬ 
ever, that this is not the case. We shall first show that 
when the bottom is acoustically uniform, so that rays 
in the bottom are straight lines, the intensity pre¬ 
dicted by ray theory for a ray such as EABH in 
Figure 23 is zero. Figure 25 shows a ray having 
inclination 0 in the water, 0 t in the bottom, together 
with a neighboring ray. By Snell’s law of refraction 
we have 

c 

cos 0i = — cos 0 (9) 

Cl 

where c and Ci are the velocities of sound in water and 
bottom respectively. Now the energy which leaves 
the source E in an interval dO of inclinations and in a 
fixed narrow interval of azimuth is partly reflected 
and partly transmitted, and the transmitted part is 
distributed over the region between the rays .IP 
and A'P' in Figure 25. If R(6) is the reflection 



Figure 25. Spreading of adjacent sound rays on enter¬ 
ing the bottom. 


As the ray AP approaches the horizontal, sin 0i ap¬ 
proaches zero, and according to equation (11) the 
intensity at P must do likewise. This conclusion is 
made even stronger by the fact that, according to 
Section 2.6.2, P(0) approaches unity as 0 approaches 
the angle for total reflection. Thus, ray theory cannot 
account for the sound received via a path like EABH 
in Figure 23, when the bottom is uniform. 

The argument just given to show the inapplicabil¬ 
ity of ray theory to arrivals of the type shown in 
Figure 23 would of course not be strictly correct if 
there were a gradual increase of the velocity of sound 
with depth in the bottom, a situation which is quite 
common, especially for soft bottoms. It will be in¬ 
structive to consider briefly the sound ray paths for 
this case, since the limitations of ray theory can be 
most clearly seen by studying this case where it is 
partially applicable. 



Figure 26. Ray paths in and over a bottom giving 
weak upward refraction. 


coefficient of the bottom at the angle 0, this trans¬ 
mitted energy is proportional to [1 — P(0)]f/0. If the 
range r = AP is large compared to EA, the distance 
between P and P' will be approximately 

c sin 0 , 

d,s ~ rdd i = r —;- dd (10) 

Ci sin 0i 


by equation (9). By introducing another factor r to 
allow for azimuthal spreading, the energy received 
at P per unit area is then proportional to 


[1 - P(0)]d0 


1 

2 


ci - Rm ■ 


Ci sin 0i 
c sin 0 


( 11 ) 


Figure 26A shows a family of rays connecting a 
source and hydrophone, both of which are lying on a 
bottom characterized by weak upward refraction. 
According to ray acoustics the first signal to reach 
the receiver H will arrive via the path I„. This will be 
followed almost immediately by arrivals along other 
paths, such as ly, which likewise lie in the bottom 
but which involve one or more reflections at the inter¬ 
face between bottom and water. Some time later an¬ 
other group of arrivals will be received, each of which 
comes along a path involving one reflection from the 
surface of the water. One path of this type is shown 


rds 


r 





























224 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


in Figure 26A and labeled II,,; many other such 
paths, not shown, are also possible; some of them in¬ 
volving additional reflections from the water-bottom 
interface as was the case for I). This second group of 
arrivals will in turn be followed by another group, 
exemplified by III e in Figure 26A, involving two re¬ 
flections from the free surface, and so on. Mixed in 
with these arrivals along paths which enter and leave 
the bottom will be those along paths lying wholly 
in the water, shown in Figure 26B. These paths have 
for simplicity been drawn for the case where the 
velocity of sound in the water is uniform. In practice 
most of the experiments performed so far have en¬ 
countered isothermal water with consequent upward 
refraction; this case, which will be discussed later, is 
in most respects little different from the uniform case 
considered here. The first arrival among these water 
rays will be along the direct path I i0 , the second along 
the surface-reflected path II W , the third along a path 
III,,, involving one reflection from the bottom, and 
so on. 

Thus if the predictions of ray acoustics were valid, 
we should expect the signal received at the hydro¬ 
phone to consist of a number of evenly spaced groups 
of pulses of diminishing strength, corresponding to 
the “ground rays” shown in Figure 26A, plus a 
number of individual pulses starting at a later time 
and separated by gradually increasing intervals, 
which correspond to the “water rays” shown in 
Figure 26B. Of these various arrivals, some are posi¬ 
tive pressure pulses, others negative, according to 
the number of phase-changing reflections each has 
suffered. 

The extent to which the predictions of ray theory 
can be trusted in a case of this sort can be estimated 
by resolving the explosive pulse into a superposition 
of sine waves by use of Fourier’s theorem, as de¬ 
scribed in Section 9.2.4, and then applying the criteria 
given at the end of Section 3.6.2 for applicability of 
the eikonal equation to sinusoidal waves. It is clear 
from these criteria that the condition for ray theory 
to be applicable to a sine wave along a path of the 
type I g , II a , etc., in Figure 26A, is that the maximum 
depth of the path, shown as dj for ray I„, should be 
large compared to the wavelength of the sound in 
the bottom. This condition will be fulfilled by the 
highest frequencies in the Fourier resolution of the 
explosive pulse, but not by the lowest frequencies; 
moreover, the frequency above which ray theory is 
applicable recedes to higher and higher values for the 
successive arrivals I„, II„, III„, etc. As is to be ex¬ 


pected, this critical frequency approaches infinity as 
the magnitude of the velocity gradient in the bottom 
decreases to zero, since the depths of penetration rf I( 
etc., of the rays approach zero. 

A similar consideration of the disturbance propa¬ 
gated through the water suggests that ray theory 
should fail for frequencies of the order of c/h and 
smaller, where h is the depth of the water. This limit 
has little meaning, since this frequency can be ex¬ 
pected to be lower than the frequency at which the 
ray picture fails for the ground rays, and we cannot 
make a clear separation between the ground dis¬ 
turbance and the water disturbance after we have 
abandoned the ray concept. 

We may thus expect the pressure variation which 
would be recorded by a very high-fidelity receiver at 
// to consist of the succession of pulses which ray 
theory would predict plus a correction which is made 
up almost entirely of low frequencies. For the dis¬ 
turbance due to the shock wave from the explosion, 
the times of the various ray arrivals can, ideally at 
least, be identified on the oscillogram of the received 
pressure by the occurrence of sharp jumps in the 
pressure; these jumps, due to the sudden rise at the 
front of the shock wave, cannot easily be obliterated 
by the low-frequency correction (see Figure 13). 
Since, as explained above, the intensities of the ar¬ 
rivals predicted by ray theory are zero for a uniform 
or downward-refraction bottom and are small for a 
bottom with weak upward refraction, we should not 
be surprised to find the disturbance received by the 
hydrophone to be dominated by the low-frequency 
portion, with only a few detectable traces of the ray 
arrivals. 

To determine the nature of the low-frequency cor¬ 
rection just mentioned, it is necessary to study solu¬ 
tions of the wave equation similar to those considered 
in Section 2.7.2. In a report prepared by CUDWR, 23 
it is shown how the normal modes of vibration of 
water and bottom can be computed and superposed 
to correspond to the disturbance produced by ex¬ 
plosive source. The mathematical details are too 
complicated to be given here; 0 however an attempt 
will be made below to explain in a simplified manner 
the physical basis for some of the most important 
results of reference 23. In particular, it will be shown 
how many characteristics of the signal received at the 

c The reader who wishes to study the mathematical theory 
of normal modes will find it profitable to study also the treat¬ 
ments devoted primarily to single-frequency sound in deep 
water 25 and electromagnetic waves in the atmosphere. 26 




BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


hydrophone can be interpreted in terms of a simple 
dispersion law, i.e., a propagation of different fre¬ 
quencies with different velocities. 

The physical reasons underlying the dispersion 
phenomena just mentioned can be seen by consider¬ 
ing the simple case of a progressive wave of a single 
frequency /. Let us try to construct such a wave by 
assuming the pressure disturbance to be 

p = e 2 * i{x/x ~ Jt) M(z) } (12) 

where x is a horizontal coordinate, and z is the depth 
below the free surface of the ocean. This function p 
must satisfy the wave equation 



in the water; that is, when z is less than the depth h 
and must satisfy the analogous wave equation 11 



in the bottom, that is, when z is greater than h. In 
addition, p must satisfy boundary conditions at the 
free surface and at the interface between water and 
bottom. These conditions are 


p = 0 at z = 0 

Pwater Pbottom 2 = h. 


/'water /'bottom " 1 

am = n dp\ 

\p dZ/water Vpi dz/ h 


at, z = h. 


(15) 

(16) 

(17) 


Assuming for simplicity that water and bottom are 
uniform, so that c and Ci are independent of z, we 
have, on inserting expression (12) into equation (13), 


d 2 M 

dz 2 



P 

<?. 


M for z < h. 


(18) 


To satisfy this equation and the condition (15), we 
must take 


M = A sin ^27r|/^ — ^ z'j for z < h. (19) 


This equation is formally correct regardless of whether 
the quantity under the radical is positive or negative. 
However, it will be shown below that this quantity 
must be positive if equation (12) is to represent a 
physically possible disturbance. Similarly, to satisfy 
the wave equation in the bottom we must have 


d The theory presented here ignores shearing stresses in the 
bottom and thus treats the bottom as a fluid rather than as a 
solid. This assumption, although reasonable for MUD bot¬ 
toms, is of course not true for ROCK. However, many features 
of the disturbance predicted by the present theory would 
doubtless also be observed over a rock bottom. 


225 


= 4.^ - ■fiilf for z> h. (20) 

oz La- CxJ 

Now, if [(1/X 2 ) — CPAi)] is negative, M will be a 
periodic function of z in the bottom, and according 
to equation (12) the pressure disturbance in the bot¬ 
tom will consist of progressive waves going diagonally 
up or down. The disturbance created by an explosion 
will consist in part of a superposition of progressive 
waves of this type which travel diagonally downward 
in the bottom; these waves are, however, a relatively 
unimportant part of the signal received in the water 
at a great distance, since their energy spreads out in a 
downward direction and thus decreases fairly rapidly 
with distance in the horizontal plane. The part of the 
signal which is most important in the present applica¬ 
tion consists, instead, of a superposition of waves of 
the form (12), for values of X and / which make 
[(1/X 2 ) — (/»/<?)] in equation (20) positive. The two 
independent solutions of equation (20) for this case 
will be exponential functions of z, one increasing to 
infinity as z increases, the other decreasing to zero. 
The former of these is physically inadmissible; so we 
may conclude that if a pressure wave of the desired 
form exists at all, it must be of the form 

M = B exp ^-2 tt|/^ - "^zj for z > h, (21) 

and of course of the form (19) for z < h. However, it 
is easily shown that no matter what values are given 
to the constants A and B, it is not possible to satisfy 
both of the boundary conditions (16) and (17) unless 
X and / are related in a particular way. For, on in¬ 
serting expressions (19) and (21) into these condi¬ 
tions, and using the abbreviations 



we obtain 

A sin ph = Be r 4 * (22) 

pA vB , 

— cos ph = - e~ vh . (23) 

p Pi 

Dividing the first of these equations by the second 
eliminates A and B, and gives the following relation 
which must be satisfied by / and X. 

-tan ph = ——• (24) 

p v 

If this relation is satisfied, a suitable choice of the 
ratio B/A will insure that both equations (22) and 
(23) are satisfied. It is easily verified that if Ci > c 









226 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


FREQUENCY IN C, fi • 100 FEET 



/ = Frequency. c = Velocity of sound in water. 

h = Depth of water. X = Horizontal wavelength of disturbance. 

Figure 27. Variation of wavelength and phase velocity with frequency for normal modes in shallow water. Velocity of 
sound in bottom assumed to be 1.5 X c. Density of bottom assumed 2 times density of water. 


and pi > p, equation (24) cannot he satisfied if p is 
imaginary; this justifies the statement made in the 
second sentence following equation (19). 

Graphs of the solutions of equation (24) are given 
in Figure 27 for a typical set of values of Ci, pi, and h. 
Typical curves of the variation of pressure along a 
vertical line are given in Figure 28, corresponding to 
particular points on the graphs of Figure 27. As was 
explained in Section 2.7.1, it is customary, by analogy 
with the terminology used in the theory of vibrating 
systems of particles, to use the term “normal mode” 


to describe a state of vibration of the water and 
bottom in which the pressure distribution is given by 
equation (12); for modes of the present type this is 
equivalent to a disturbance of the type shown in 
Figure 28 and having an amplitude represented by a 
horizontally moving sine wave. It is convenient to 
identify families of these normal modes by the num¬ 
ber of horizontal planes in the water, including the 
free surface on which the pressure is always zero. 
This number is called the order of the normal mode, 
and is indicated by the labels “FIRST MODE,” 



















BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


227 


M(z) ARBITRARY scale 



c 


SECOND MODE — = 1.5 


MU) ARBITRARY SCALE 



1.5 


MINIMUM FREQUENCY 
FIRST MODE 
MU) ARBITRARY SCALE 



THIRD MODE ^'2.0 


/ = Frequency. 
h = Depth of water, 
c = Velocity of sound in water, 
z = Distance below surface of water. 

M(z) = Pressure amplitude at depth z. 

Figure 28. Variation of pressure with depth along a 
vertical line for various normal modes. Velocity of 
sound in bottom assumed to be 1.5 X c. Density of bot¬ 
tom assumed 2 times density of water. 


‘‘SECOND MODE,” and so on, in Figures 27 and 28. 

A noteworthy fact is that for modes of any given 
order there is a minimum frequency below which no 
value of X can be found which will satisfy the 
boundary conditions. At this frequency the quantity 
v, which is inversely proportional to the depth of 
penetration of the disturbance into the bottom, goes 
to zero; and the pressure distribution takes a form 
such as that shown in Figure 28B. As the mathemati¬ 
cally inclined reader can verify for himself from equa¬ 
tion (24), the minimum frequency for the first mode 
has a half-period equal to the interval which ray 
theory would predict between the arrivals of types 
l g , l\g, Ill fl , etc., of Figure 26. This half-period is 
given by 



Since these ray arrivals are alternately positive and 
negative, the period of the disturbance given by ray 
theory is the same as that for the minimum frequency. 
It is also noteworthy that the minimum frequency 
for the idh mode is (2v — 1) times its value for the first 
mode. This has the very important consequence that 
any simple harmonic disturbance of low frequency 
which is propagated over a large horizontal range can 
be represented by a superposition of a finite number 
of normal modes of low order. 

Let us now consider the velocity of propagation of a 
disturbance which consists of a superposition of nor¬ 
mal modes of a given order but distributed over a 
narrow range of frequencies. It is easy to show that 
such a disturbance, considered as a function of hori¬ 
zontal distance x or of time t, will form a wave train. 
For each component normal mode has a phase factor 
proportional to e 2 ’ ri ( x / x-/< ). At any given time t 
there will be some value of x for which most of the 
component modes are approximately in phase; in the 
neighborhood of this value of x the pressure disturb¬ 
ance will therefore be large. If x differs very widely 
from this value, on the other hand, the phases of all 
the component normal modes will be rather randomly 
distributed since the different modes have slightly 
different wavelengths X; for such values of x the 
pressure disturbance will be small. If we watch the 
motion of the wave train in the course of time, we 
shall find that the region of large amplitude moves 
with a certain velocity, commonly called the “group 
velocity.” Now, if the center of the wave train is to 
be near x x at time t\, and near x 2 at time t 2 , the phase 
change ( x 2 — Xx/\) — f(t 2 — h) must be very nearly 
the same for all the different normal modes contained 
in the wave train, in order that they may continue 
to reinforce one another. This implies, in the limit 
where only a very narrow range of frequencies is in¬ 
volved, 


dfl 


x 2 — Xx 


- m - h)] = o, 


or, if V is the group velocity, 


(x 2 - x Y ) 
(4 — tx) 



(26) 


Now the phase velocity of any single-frequency 
component, defined as the speed of advance of a 
point having a given constant value of the phase 
2ir(x/\ — ft), is equal to X/. If this quantity were a 
constant independent of frequency, as is the case for 































228 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


sound propagated in a single homogeneous medium, 
the expression (26) would simply equal the phase 
velocity. For the disturbances we are considering 
here, however, X/ is not independent of frequency, as 
a glance at Figure 27 will show. 

The importance of the result (26) in shallow-water 
transmission is that it enables us to understand the 
dispersion phenomena in the ground and water 
waves. The initial disturbance can be represented as 
a superposition of normal modes having a very wide 
range of frequencies. However, since the group 
velocity is different for different frequencies, the dif¬ 
ferent frequencies in this superposition will get sepa- 



C 


Figure 29. Dispersion in group velocities of normal 
modes. Frequency in /; depth of water, h ; velocity 
of sound in water, c. 

rated out somewhat at long ranges, and each band 
of normal modes of a given order and a given narrow 
range of frequencies will be propagated with its own 
group velocity. This effect is shown quantitatively 
for a typical set of conditions in Figure 29. The curves 
of this figure are derived from those of Figure 27 by 
differentiation. Note that the group velocity varies 
from the ground velocity Ci at the low cutoff fre¬ 
quency to the water velocity c at very high fre¬ 
quencies, but has a minimum at an intermediate 
frequency. The existence of this minimum produces 
an interesting effect, which will be described later. 

The main features of the disturbance received at a 
distance from an explosive source can be explained 
most simply by concentrating attention on one of 
these curves, say that for the first mode. This will not 
only be illustrative of the main characteristics shared 
by all the normal modes, but will in fact provide a 
rough prediction of what some of the actual records 
to be discussed in Section 9.4.3 should look like. For 
it has been pointed out that there exists a minimum 


frequency for each normal mode and that the fre¬ 
quency for the first mode is the lowest. Thus if the 
disturbance produced by an explosion is received with 
equipment responsive only to sufficiently low fre¬ 
quencies, the resulting signal can be interpreted in 
terms of the first-order modes alone. Even when high- 
fidelity recording equipment is used, the first mode 
should dominate the initial or ground wave portion 
of the disturbance, since it can be shown theoretically 
that the amplitude of the first mode is greater than 
the amplitude of higher modes in this region. 24 

Let us therefore suppose that we have a source of 
sound which generates a transient disturbance con¬ 
sisting entirely of a superposition of first-mode vibra¬ 
tions of various frequencies. Since according to 
Figure 29, the highest group velocity occurs for the 
lowest frequencies above the cutoff, the first sound to 
arrive at a distant hydrophone will be a wave train 
whose frequency corresponds very nearly to point A 
of Figure 29. The disturbance arriving a little later 
will consist of frequencies having a slightly slower 
group velocity, that is, of slightly higher frequencies. 
Thus, the frequency of the received disturbance will 
gradually increase with time until the value corre¬ 
sponding to point B is reached. At this moment, the 
very highest frequencies present in the original dis¬ 
turbance start to come in, traveling in the limit with 
the velocity c. From this time onward, the received 
disturbance consists of a low-frequency part and a 
high-frequency part superposed, the former con¬ 
tinuously increasing its frequency along the branch 
BC of the dispersion curve, and the latter continu¬ 
ously decreasing its frequency along the branch FG. 
Eventually these two coalesce, and the disturbance 
dies out at an intermediate frequency. 

All these characteristics are apparent in the theo¬ 
retical pressure-time curve of Figure 30, which shows 
the contribution of the first mode to the disturbance 
produced under a typical set of conditions by a source 
which emits a single positive-pressure pulse of short 
duration. The portions of the curve corresponding 
to the points A, B, C, F, G of Figure 29 are labeled 
with these letters. Similar curves showing the con¬ 
tributions of normal modes of higher order are given 
in reference 23. These have lower amplitudes than 
that for the first mode, especially during the “ground 
wave” phase, that is, the portion of the disturbance 
which has traveled with a velocity greater than c and 
thus lies to the left of B and F. According to the 
present theory, which idealizes the bottom as a 
homogeneous fluid, the variation of the pressure at 























BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


229 



o 


0.05 0.10 0.15 


0.20 0.25 0.30 


uj TIME FROM BEGINNING OF GROUND WAVE, IN SECONDS 

< 

o 



Figure 30. Theoretical contribution of the first mode to the disturbance at a distance from an explosion in shallow water. 
Source and receiver both assumed to be on the bottom. Range: 9,200 yards. Depth of water: 60 feet. Velocity of 
sound in bottom = 1.1 X velocity in water. Density of bottom = 2 X density of water. (Vote, A should appear at point 
indicating beginning of ground wave.) 


the hydrophone with time should be given by the 
sum of these contributions from all the normal modes, 
plus certain additional terms whose magnitude de¬ 
creases rapidly with increasing range, so that they 
become negligible at very long ranges. This complete 
pressure-time curve would of course show sharp 
jumps at the positions corresponding to the arrivals 
predicted by the ray picture. 

In the following section we shall compare these 
theoretical predictions with observations. In this 
comparison certain factors have to be taken into 
consideration which for simplicity have been neg¬ 
lected in this section, such as the modification of the 
received disturbance by the frequency response char¬ 
acteristics of the recording equipment, and the fact 
that instead of delivering a single impulse, an ex¬ 
plosion gives out a shock wave followed by several 
bubble pulses (see Section 8.6.). 

9.4.3 Analysis of Experimental 
Records 

The Woods Hole Oceanographic Institution has 
obtained a large number of oscillographic records of 


sound from explosions in shallow water at distances 
between 0.25 mile and 30 miles. 20 Several series of 
experiments were conducted at widely separated 
places with bottoms of mud, sand, and coral. The 
depths of the water at the sending and receiving 
positions were usually similar and in the range 40 to 
180 ft; some shots were made at greater depths. 
The hydrophones used were in all cases placed on the 
bottom, while the charges were usually on the bottom 
but sometimes at mid-depth. Charges of J^-lb TNT 
to 300-lb TNT were used. At all stations the water 
was very nearly isothermal, so that, sound rays in the 
water were refracted slightly upward. 

Figure 31 shows some typical oscillograms of the 
sound received in these experiments. Each record 
consists of eight traces simultaneously recorded. The 
first of these, labeled “time break,” is used merely to 
record the instant at which the charge was set off; the 
others record the disturbance received, as modified 
by the frequency responses shown at the left for the 
various recording channels. Most of the interesting 
features show up best on the two channels labeled 
“Mark II low frequency,” which record the same 























































230 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 


f RADIO SIGNAL 

2 MARK 31 HIGH FREQUENCY 

3 GEOPHONE 

4 MARK I■ 

5 RADIO BUOY GEOPHONE 

6 MARK 31 LOW FREQUENCY 


7 MARK 31 RECTIFIED 



IN C 


DEPTH OF WATER 90 FEET 
RANGE 3900 YARDS 


I RADIO SIGNAL 

2' MARK I HIGH FREQUENCY 

3' GEOPHONE 

4' MARK 3E RECTIFIED 

5" MARK I 

6' MARK 3E LOW FREQUENCY 
Y MARK IE RECTIFIED 



2 3 4 

ie io io io 

FREQUENCY 
IN C 


SHOT 95, NEAR JACKSONVILLE, FLORIDA 
CHARGE: 25 LBS TNT ON BOTTOM 
DEPTH OF WATER 60 FEET 
RANGE 4200 YARDS 


Figure 31. Typical records of explosive sound transmissions in shallow water. Times marked along the top of each 
oscillogram are in seconds. Curves at left give relative amplitude response of each channel to the various frequencies. 


signal at two different amplitude levels. However, the 
rectified traces show most clearly the times of the 
various water wave arrivals, namely, shock wave and 
bubble pulses. The arrival time of the first of these is 
especially useful, since the range can be determined 
for any shot by multiplying by c the interval between 
the detonation and this arrival. 

The interpretation of records like those of Figure 
31 is often complicated by the fact that each ob¬ 
served trace represents a superposition of the dis¬ 
turbances produced by the shock wave and all the 
bubble pulses. According to the theory of normal 
modes, the amplitude of the disturbance produced 
by any one such pulse of very short duration should 
be proportional to the impulse f pdt of the pulse; since 
this quantity is of the same order of magnitude for 
the shock wave and the first one or two bubble waves, 
the resulting superposition can become very compli¬ 


cated. However, as was mentioned in Section 8.6, the 
bubble pulses are often much weaker when, as was 
usually the case in the WHOI experiments, the charge 
is fired in contact with the bottom. Records (A) and 
(B) of Figure 31 are fairly typical examples of shots 
on the bottom; the former shows a strong bubble 
pulse and the latter a very weak one. Note that the 
separation of the first two high peaks in the ground 
wave of Figure 31A is just equal to the bubble period 
as read from the rectified trace. Since the periods of 
the oscillations are long compared with the duration 
of the impulse sent out by the explosions, the only 
noticeable effects of increasing the size of the charge 
are to increase the amplitude and to alter the time 
lag in the arrival of the bubble pulse effects. Chang¬ 
ing the position of the charge from bottom to mid¬ 
depth also seems to have very little effect. 

Let us begin the detailed discussion of the ob- 

























































BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


231 


RANGE IN YARCS 



Figure 32. Typical plot of travel time against range for components of different frequencies in the ground wave. Loca¬ 
tion, near Jacksonville, Florida, at mean depth of 57 feet, charge and hydrophone both resting on bottom. 


served records by considering the initial or ground 
wave phase of the disturbance. All the records agree 
in a general way with the predictions of the theory 
of reference 23 as outlined in Section 9.4.2 (see 
Figure 30) in showing a gradual increase of frequency 
between the beginning of the disturbance and the ar¬ 
rival of the water wave. According to equation (25) 
of Section 9.4.2, when the bottom is uniform the 
period of the disturbance at its beginning is a func¬ 
tion of the velocity Ci of sound in the bottom. Exten¬ 
sion of the theory to cases where the bottom con¬ 
sists of a deep firm stratum overlaid by a slower one 
gives the result that at sufficiently long ranges the be¬ 
ginning of the ground wave should have a frequency 
dependent in a complicated way upon the velocities 
in both layers, but that, if the upper layer is suf¬ 
ficiently thick in comparison with the depth of the 
water, a strong new disturbance of distinctly higher 
frequency will arrive some time later, the arrival 


time and frequency of this new disturbance being 
approximately the same as for the ground wave which 
would occur if the upper layer were infinitely thick. 
These theoretical predictions suggest that noting the 
frequencies of the first arrival and any subsequent 
arrivals in the ground wave may provide useful in¬ 
formation about the different strata in the bottom. 
This is illustrated in Figure 32, which may be com¬ 
pared with Figure 24. Here many of the records show 
a recognizable new arrival of different frequency from 
the first which comes some time later. Complete in¬ 
terpretation of the data shown in Figure 32 is diffi¬ 
cult, but there is definite evidence for a layer in which 
the velocity of sound is about 1.5 times the velocity 
in water, as well as of one or two layers of higher 
velocity which determine the times of the first ar¬ 
rivals. It is noteworthy that this dependence of the 
frequency of a ground wave arrival upon the velocity 
of sound in the layer chiefly responsible for the ar- 
































232 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 




Figure 33. Dispersion in the water wave produced by an explosion in shallow water. Shot 90, near Jacksonville, Fla.; 
charge, 5 lb TNT on bottom; mean depth of water, 57 ft; range, 7,000 yd; frequency response of channels as shown in 
Figure 31. 


rival in question can be observed even in the first 
arrivals at fairly short range. Thus in the data from 
which Figure 24 was constructed, the period of the 
first arrival was between 0.024 and 0.036 sec for the 
points to the left of the intersection of the two 
straight lines, and was between 0.050 and 10 sec for 
the points to the right, except for two very close to 
the intersection. 

On some records the ground wave dies out quite 
noticeably before the arrival of the water wave. The 
theorjr of reference 23 indicates that if the bottom is 
uniform to all depths, the amplitude of the ground 
wave should increase steadily until the water wave 
arrives, and that a decrease in the strength of the 
ground wave in this region implies the presence of 
layers of different materials. In the latter case, there 
may be a secondary ground wave arrival of the sort 
mentioned in the preceding paragraph, which dies 


out considerably before the arrival of the water wave. 

Let us now consider the disturbance after the arrival 
of the water wave. Figure 33 shows, in more detail 
than Figure 31, the dispersion phenomena occurring 
in this stage. The third trace from the bottom shows 
most clearly the ground wave just before the arrival 
of the water wave, and the gradual development of 
the water wave from a disturbance of very low ampli¬ 
tude and high frequency, superposed on the ground 
wave, to the final, so-called Airey phase where 
ground wave and water wave fuse at an intermediate 
frequency and die out. The similarity of this record 
to Figure 30 is quite striking. The resemblance is not 
nearly so close for the second trace from the top, 
since this trace was recorded with more fidelity at the 
high frequencies, so that many normal modes higher 
than the first contribute significantly to it. 

About 0.2 sec after the main disturbance has died 









BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


233 



h — Depth of water. A z = Depth below the surface of the bottom above 

/ = Frequency of contribution of first normal mode. which 99% of the wave energy of the first mode in 

c = Velocity of sound in water. the bottom is included, 

ci = Velocity of sound in bottom, assumed uniform. 

Figure 34. Typical curves of the frequency dependence of the depth of penetration of the first normal mode into the bot¬ 
tom. Density of bottom assumed 2 times density of water. 

out, another disturbance is recorded, weaker than the 
first and due to the bubble pulse. On the low-fre¬ 
quency trace this second water wave looks similar to 
the first; but on the high-frequency trace it is very 
different. Frequencies above about 200 cycles are 
absent in the bubble pulse disturbance but strong 
in the primary disturbance. This is probably due to 
the fact that, as indicated in Figure 8 of Chapter 8, 


the pressure delivered by the bubble in its contracted 
stage has a duration of several milliseconds and is 
thus lacking in high-frequency components. 

A detailed analysis of the dispersion phenomena in 
the water wave can provide useful information on the 
characteristics of the upper layers of the bottom. 23 
Unless shots are made at very short ranges, the arrival 
times and frequencies of ground waves furnish inf or- 














234 


TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



h = Depth of water. 

/ = Frequency of contribution of first normal mode. 
t = Time between explosion and arrival of frequency/. 
to = 


- Theoretical dispersion in the first normal 

mode for various uniform bottoms all of den¬ 
sity 2. 

-— Same for layer of thickness O.lh and C\/c = 1.1 

underlain by infinite layer with c«/c = 3, both 
of density 2. 

-Same for layer of thickness h and C\/c = 1.1 

underlain by infinite layer with o>Jc = 3, both 
of density 2. 

Figure 35. Theoretical and observed dispersion in the water wave. Shots were made off Jacksonville, Fla., where the 
depth of water was 115 to 120 feet. Hydrophone was on the bottom for all shots. Observed frequencies are taken from the 
Mark II low-frequency record whose response is shown in Figures 31 and 33. 


Time between explosion and first water wave ar¬ 
rival. 

Group velocity for frequency /. 

Velocity of sound in water. 

Ci, c-i = Velocities of sound in bottom lavers. 


V = 
c = 


mation only on those layers of the bottom which are 
reasonably thick, in comparison with the depth of the 
water; the water wave, on the other hand, can supply 
information on the uppermost layers even when they 
are much thinner. This is a consequence of the fact 
that for normal modes of any order the higher the 
frequency the more rapidly the disturbance dies out 


with increasing depth. Since the frequencies in the 
water wave are much higher than those in the ground 
wave, the water wave will not penetrate so deeply 
into the bottom, and will thus be less affected by the 
characteristics of deep layers and more affected by 
the characteristics of the top layers. Figure 34 shows 
how the depth of penetration varies with frequency 


















































BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 


235 


for the first mode for various types of bottoms. 
Figure 35 shows the theoretical dependence of the 
frequency of the first mode on the time, for these 
same types of bottoms, and shows also the observed 
frequencies as recorded in a particular region on the 
low-frequency Mark II channel, whose frequency 
response has been shown in Figures 31 and 33. Be¬ 
cause of its suppression of high frequencies, the record 
of this channel should consist principally of the con¬ 
tribution of the first mode. It will be noticed that at 
the highest frequencies the slope of the theoretical 
frequency-time curve is nearly independent of the 
assumed structure of the bottom, and that the ob¬ 
served points show the same slope. Note that the 
frequency corresponding to a given group velocity 
over a uniform bottom varies inversely as the depth 
of the water, a relation which is easily verified from 
equation (24). This relation is fairly well confirmed 
experimentally. 

The observed points in Figure 35 do not follow 
any of the curves for a uniform bottom but indicate 
rather that the velocity of sound increases with 
depth. A rough estimate of the scale and nature of 
this increase can be obtained by studying Figures 34 
and 35 together. Thus at higher frequencies than 
that corresponding to fh/c equal to 4.5, the points of 
Figure 35 scatter evenly about the curve for C\/c 
equal to 1.1. According to Figure 34, for fh/c equal 
to 4.5, 99 per cent of the wave energy in the bottom 
is confined within a layer of thickness 0.2 times the 
depth of the water. We may therefore say that the 
velocity of sound in the bottom is 1.1c down to a 
depth of the order of 20 ft. A similar readingof thetwo 
figures at fh/c equal to 1.5 gives the result that some 
sort of weighted average of the velocities over a depth 
in the bottom of the order of half the depth of the 
water has a value intermediate between 1.3c and 
1.5c. These rough conclusions are confirmed by 
the fact that the points of Figure 35 fall between 
the dotted and dashed curves. Information ob¬ 
tained from study of the ground waves regarding 
the deeper layers of the bottom should, of course, 
be borne in mind when studying the water waves 
in this way. 

Measurements have been made on the maximum 


intensity of the water wave as recorded by the various 
channels. These indicate a decrease of the recorded 
maximum intensity with distance, which in different 
regions varies from an inverse fourth or fifth power to 
an inverse 2.4 power. Theoretically, if there is no 
absorption or scattering, the energy in any normal 
mode should vary inversely as the first power of the 
distance, because the normal modes spread out hori¬ 
zontally but not vertically. Since the duration of the 
signal increases proportionally to the range, the peak 
intensity of any normal mode should decrease about 
as the inverse square of the range; the more refined 
calculations of reference 24 show an inverse 5/3 power 
dependence. Thus, although the experimental data 
are scattered and hard to interpret, there appears to 
be a discrepancy between theory and experiment. 
One would, of course, not expect perfect agreement 
with a theory which neglects absorption and scat¬ 
tering in the bottom, especially since most of the ex¬ 
periments were conducted over soft bottoms. 

At one of the stations where shots were made, 
near the Orinoco delta, it was found that no fre¬ 
quencies below about 300 c appeared in the water 
wave, although a normal dispersion record was ob¬ 
served in deeper water in the same locality. The 
ground wave on the anomalous records was fairly 
normal. 

A few shots were made near the Virgin Islands 
with land between the shot and the hydrophone. 
These showed a ground wave similar to that which 
would have been observed in the absence of the land, 
but the water waves were entirely absent. A related 
observation is that blasting explosions on land gave 
weak ground waves at a hydrophone in the sea off¬ 
shore, but no water waves. 

Shots made near Solomons, Md., produced low- 
frequency disturbances of periods from 0.1 to 0.3 sec 
which were propagated with a low velocity, about 
1,700 ft per sec. These disturbances have been tenta¬ 
tively ascribed to the so-called Rayleigh wave, which 
is a surface-bound wave in the bottom whose propa¬ 
gation involves shearing stresses. Such waves fall 
outside the province of the theory of the preceding 
Section 9.4.2, which idealizes the bottom as a fluid 
medium. 




Chapter 10 

SUMMARY 


R esearch on sound transmission during World 
War II was concerned almost exclusively with 
the investigation of sound fields which were operation¬ 
ally important. More than half of the experimental 
work was devoted to the sound field of standard echo- 
ranging transducers operating at frequencies around 
24 kc. The purpose of the work was primarily to pro¬ 
vide information which could be used to increase the 
effectiveness of Navy gear already in use on sub¬ 
marines and antisubmarine vessels. The instrumen¬ 
tation used for research differed as little as possible 
from standard operational gear; what modifications 
were made usually represented the minimum neces¬ 
sary for quantitative evaluation of the data obtained. 
Questions which did not seem important opera¬ 
tionally, such as the physical cause of the observed 
attenuation of supersonic sound in the sea, received 
scant attention in these studies. 

In the sections which follow, the essential results 
of these experiments on underwater sound trans¬ 
mission are summarized. Section 10.1 lists the defi¬ 
nitions of the most important quantities used in 
describing underwater sound fields. Sections 10.2 and 
10.3 summarize what is known concerning the aver¬ 
age transmission of supersonic and sonic sound in the 
sea. In Section 10.4, data on the fluctuat ion and varia¬ 
tion of seaborne sound are summarized. Finally, 
Section 10.5 provides a brief discussion of probable 
trends in future research on sound transmission. 

10.1 BASIC DEFINITIONS 

lo.i.i Sound Pressure and Sound 
Field Intensity 

A sound wave in a fluid can be described con¬ 
veniently in terms of the pressure disturbance which 
arises in the vicinity of a sound source, travels 
through the fluid, and is finally received by a hydro¬ 
phone. The instantaneous sound pressure is the dif¬ 
ference between the instantaneous value of the pres¬ 
sure at a chosen location and the mean or equilibrium 


pressure at the same point. The rms value of the 
instantaneous sound pressure is usually called the 
rms sound pressure. Usually, the average is carried 
out over a time interval which is long compared with 
the periods of the principal frequencies making up the 
sound signal. In the case of single-frequency sound, 
the average is extended over one period (or an inte¬ 
gral number of periods). Unless specified otherwise, 
“sound pressure” as used in the technical literature 
is short for rms sound pressure. Except in the case 
of standing waves, the rms sound pressure is an ex¬ 
cellent measure of the energy carried by the sound 
wave. At the present time, sound pressure values 
are uniformly reported in units of dynes per square 
centimeter. 

The sound field intensity is defined as the averaged 
power carried by a sound wave per unit cross section 
of a wave front. The units in present use are watts 
per square centimeter. If the radii of curvature of 
the wave fronts are large compared with the wave¬ 
length, then the rms sound pressure and the sound 
field intensity are connected in excellent approxima¬ 
tion by the formula 

I = 10 - 7 -, ( 1 ) 

pc 

in which p is the rms sound pressure, p is the density 
of the fluid in grams per cubic centimeter, c is the 
sound velocity in the fluid in centimeters per second, 
and I is the sound field intensity. 

10.1.2 Sound Level 

The sound field intensity is usually reported on a 
logarithmic scale. The most common scale for this 
purpose is the decibel scale. The quantity L, 

L = 20 log p (2) 

in which the rms sound pressure p is expressed in 
units of dynes per square centimeter, is called the 
sound pressure level or simply the sound level. As de¬ 
fined by equation (2), L is the sound level in decibels 
above a standard which corresponds to a sound pres- 


236 


BASIC DEFINITIONS 


237 


sure of 1 dyne per sq cm. In the past, sound levels 
were frequently reported in decibels above 0.0002 
dyne per sq cm. 

10.1.3 Source Level 

The source level is a measure of the power output 
of a sound source on the decibel scale. Briefly, it is 
the sound level due to a point source at a distance of 
1 yd, in decibels above 1 dyne per sq cm. If a point 
source is located in a homogeneous, nondissipative 
medium which is infinitely extended in all directions, 
the intensity of the sound field is inversely propor¬ 
tional to the square of the distance from the source, 

F 

i = v (3) 

r- 

This law is called the inverse square law. In terms of 
the sound level, equation (3) becomes 

L = S — 20 log r. (4) 

In these equations, F and S are constants which de¬ 
pend on the power output of the source, and r de¬ 
notes the distance (slant range) from the source. 
That S is the source level as defined above can be 
verified by setting r equal to 1 in equation (4). 

For real sound sources in real media, equations (3) 
and (4) are not everywhere valid. Because of the 
finite extension of an actual sound source, the in¬ 
verse square law fails at ranges of the order of the 
dimensions of the source. Because of absorption of 
the sound in the medium and because of scattering 
and reflection from bounding surfaces, it fails at very 
long ranges. However, there is frequently an inter¬ 
mediate range interval for which equation (4) holds. 
If there is such an interval, then the constant S 
is considered the source level, even though S may 
not be the actual sound level at a distance of 1 yd. 

For a highly directional sound source, such as a 
standard echo-ranging transducer, the definition of 
the source level is further specified by the condition 
that the sound measurements are to be carried out 
on the axis, that is, the radial line of greatest sound 
field intensity. 

10.1.4 Transmission Loss and 
Transmission Anomaly 

The transmission loss H at the range r is defined 
by the formula 

(5) 


where S is the source level, and L is the sound level 
defined by equation (2). The transmission loss de¬ 
fined in this way measures the drop of the sound 
level with increasing distance from the source and 
has the virtue of being independent of the particular 
power output of the source. Other parameters of the 
source, such as operating frequency and directivity 
pattern, are known to affect the value of the func¬ 
tion H(r). The units of H are decibels. 

The transmission anomaly A is the deviation of the 
transmission loss from that functional behavior de¬ 
manded by the inverse square law of spreading. The 
defining equation for A(r) is 

d.(7-) = H{r) — 20 log r = S — L(r) — 20 log r. (6) 

The transmission anomaly vanishes if the inverse 
square law of spreading is satisfied, and it is positive 
if the sound level drops off more rapidly than 20 log r. 
Large positive transmission anomalies, therefore, 
correspond to poor sound conditions. 

In sound transmission work, it has been customary 
to train the projector in a horizontal plane on the re¬ 
ceiving hydrophone, but not to tilt the acoustic axis 
away from the horizontal. Hence, measured trans¬ 
mission anomalies will be large for a close deep 
hydrophone beneath the sound beam. 

In supersonic transmission work, it has been found 
that when successive signals are transmitted a few 
seconds apart over the same transmission path, the 
received sound intensity is subject to irregular fluctu¬ 
ations. Reported transmission anomalies always rep¬ 
resent values which have been obtained by averaging 
over a number of signals received during a brief 
period so that much of this fluctuation is smoothed 
out. 

10.1.5 Variance of Amplitudes 

The standard deviation of the individual pressure 
amplitudes in a sample of signals, divided by the 
average pressure amplitude for the sample, is called 
the variance of amplitudes for the sample. This 
variance is used as a measure of the fluctuation of re¬ 
ceived sound intensity. Observed values of the vari¬ 
ance are summarized in Sections 10.4.1 and 10.4.2. 

10.1.6 Deep and Shallow Water 

Water is effectively deep when bottom-reflected 
sound is much weaker than the direct sound; other¬ 
wise, the water is effectively shallow. Over the con¬ 
tinental shelf (depth less than 100 fathoms) the 


H{r) = S - L(r), 



238 


SUMMARY 


water is effectively shallow for most situations. Away 
from the continental shelf, the ocean is always deep 
when sharply directional sound is used (as in echo¬ 
ranging at supersonic frequencies), but may be 
shallow when listening at audible frequencies to a 
target at long range. 

10.2 DEEP-WATER TRANSMISSION 

The transmission loss in the open ocean depends 
on the way the velocity of sound changes with posi¬ 
tion in the sea, since velocity gradients distort the 
sound beam. These velocity gradients change with 
time and location, but in any localized region at any 
given time depend primarily on depth and relatively 
little on horizontal position within that region. 
Changes in sound velocity in deep water closely 
follow changes in water temperature; the effect of 
pressure changes is relatively slight and usually need 
not be considered except for transmission to great 
depths. 

The following subsections tell of the transmission 
anomalies expected for various common temperature- 
depth distributions in the ocean. 


Absorption has little effect on sound transmission 
at frequencies below 2,000 c. 

High Sonic and Supersonic Frequencies 
At frequencies above 2,000 c, the value of y a to be 
used in equation (7) is seldom greater than 0.5 in the 
open sea and is frequently so small that image inter¬ 
ference can scarcely be said to exist. Absorption plays 
an increasingly important role as the frequency in¬ 
creases. The transmission anomaly A may be com¬ 
puted from the relation. 


where r is the range in yards and where a is the at¬ 
tenuation coefficient in decibels per kiloyard. Average 
values of a at a number of frequencies are given in 
Table 2. At frequencies above 1,000 kc, the attenua- 


Table 2. Attenuation coefficient in the sea. 


Frequency 
in kc 

20 24 

30 40 

50 

00 

80 

100 

500 

1,000 

a in db per 
kiloyard 

3 4 

6 10 

13 

18 

26 

35 

150 

300 


10 . 2.1 Isothermal Water 

When the top 50 ft of the ocean are isothermal, 
transmission anomalies are determined by tw r o major 
effects, absorption and surface reflection. 

Sonic Frequencies 

At low sonic frequencies, sound is reflected from 
the sea surface in somewhat the same way as from 
a flat, perfectly reflecting mirror. The partial can¬ 
cellation of direct and surface-reflected sound reduces 
the sound intensity at long range near the surface. 
The transmission anomaly at any range may be com¬ 
puted from the equation 

( 4r/ii/i 2 0 \ 

1 - 2yaCos-^- + 7 1 ), (7) 

where hi is the depth of the sound source, ho is the 
depth of the receiving hydrophone, R is the range 
from source to hydrophone, and X the wavelength. 
The quantity y a , called the effective reflection co¬ 
efficient of the surface, is a semi-empirical param¬ 
eter; its average value for different frequencies is 
given in Table 1. 


Table 1. Effective reflection coefficient of the surface. 


Frequency in cycles 

200 

600 

1,800 

ya 

0.8 

0.7 

0.5 


tion coefficient is about three times the value pre¬ 
dicted from the viscosity of the water. At frequencies 
of 24 kc and below, a is more nearly 100 times this 
theoretical value. 


10 . 2.2 Thermocline below Isothermal 
Layer 


When sound from an isothermal layer passes at 
grazing angle into a thermocline or temperature 
layer, where the temperature decreases sharply with 
increasing depth, the sound rays are bent downward 
and become more spread out. The increased distance 
between sound rays in and below the thermocline 
reduces the sound intensity; this phenomenon is 
known as layer effect. The transmission anomaly 
below the thermocline, at ranges out to 4,000 yd, 
may be computed from the equation 


A = 5 log 


( 2Ac — r-\ 

v + ^r) 


+ 


1,000 


(9) 


where Ac is the change in sound velocity in the top 
30 ft of the thermocline. (If several thermoclines lie 
above the hydrophone or if the gradient in the ther¬ 
mocline increases with depth, Ac is the velocity 
change in the 30-ft interval giving the maximum 
value of Ac/ hi.) c 0 is the sound velocity in the surface 














DEEP-WATER TRANSMISSION 


239 


layer; h\ is the height of the sound projector above 
the top of the thermocline, that is, above the top of 
the 30-ft interval in which Ac is measured; r is the 
range from projector to hydrophone. The last term 
on the right is taken over from equation (8); the 
values of the attenuation coefficient used are given 
in Table 2. Equation (9) has been checked in detail 
at 24 kc only, but presumably gives an approximate 
indication of the anomalies expected below the 
thermocline at all frequencies above a few hundred 
cycles. At increasing depths below the thermocline, 
the anomaly decreases somewhat, the decrease being 
most marked for the shallower thermoclines. 

10.2.3 Temperature Gradients near 
Surface 

When temperature gradients are present in the 
top 50 ft of the ocean, the transmission loss from a 
projector at 16 ft to a distant hydrophone is corre¬ 
lated with the following variables: the sharpness and 
depth of the gradients (for practical purposes, the 
decrease of temperature from the surface down to 
30 ft); and D 2) the depth at which the temperature is 
0.3 F less than the surface temperature. For a deep 
hydrophone, the temperature gradients at intermedi¬ 
ate depths are also of importance. 

Sharp Surface Gradients 

When the temperature change in the top 30 ft is 
more than 1/100 times the surface temperature, the 
sound beam is bent downward by the decrease of 
sound velocity with increasing depth. The plot of 
transmission anomaly against range usually shows 
three different regions as follows: 

1. The direct sound field from the projector out to 
the shadow boundary. The anomaly within the direct 
sound field is primarily the result of absorption, and 
equation (8) is applicable. 

2. The near shadow zone. Beyond the shadow 
boundary, the sound intensity decreases very rapidly 
for some distance. Representative values for this de¬ 
crease are 50 db per kyd at 25 kc and about one- 
third this at 5 kc. These coefficients of attenuation in 
the shadow zone are apparently about half the values 
estimated from the theory of diffraction by a smooth 
velocity gradient. The range to the shadow boundary 
increases with depth in accordance with ray theory, 
but seems to be systematically somewhat less than 
predicted. 

3. The far shadow zone. With standard echo-rang¬ 
ing gear and pulses 100 msec long, the transmission 


anomaly of scattered sound at ranges of several 
thousand yards is about 50 db. Thus when the trans¬ 
mission anomaly of the direct or diffracted sound in 
the shadow zone exceeds about 50 db, the observed 
sound is scattered sound, with an anomaly which 
does not depend strongly on further increases in 
range. This scattered sound is incoherent. For short 
pulses the intensity of this scattered sound is propor¬ 
tional to the pulse length; it becomes negligibly small 
for explosive sound. 

To predict the anomalies expected under given 
temperature conditions, it is simplest to use curves 
of average anomalies for such conditions. Since un¬ 
explained deviations are frequently found between 
individual anomalies and the predictions of ray 
theory, use of average curves gives results about as 
accurate as the more elaborate methods. An example 
of this approach is Figure 40 of Chapter 5, where 
average curves are given for different values of D 2 , 
the depth at which the temperature is 0.3 F less than 
the surface temperature. 

Weak Surface Gradients 

When the temperature change in the top 30 ft is 
less than 1/100 of the surface temperature, but gradi¬ 
ents are present in the top 50 ft, the division of the 
sound field into the three regions described previously 
is usually not observed. Since a small change in such 
temperature conditions may lead to a large change of 
transmission anomaly, the observed anomalies are 
highly variable and can neither be compared with 
theory nor predicted practically with much accuracy. 
Average anomalies for different values of D 2 are given 
in Figure 49 in Chapter 5 for a shallow hydrophone. 
For a deep hydrophone, below the thermocline, equa¬ 
tion (9) may be used for approximate results. 

10 . 2.4 Sound Channels 

When the velocity of sound above and below the 
sound source is appreciably greater than the velocity 
at the source, the sound rays which leave the source 
with small inclinations will propagate out indefinitely 
without surface or bottom reflections, bending back 
and forth but always remaining within some fixed in¬ 
terval of depths. 

Surface Sound Channels 

When the sound projector lies below a sharp nega¬ 
tive gradient and above a sharp positive gradient, 
sound channel effects should be marked, with regions 
of alternately high and low anomaly found out to 



240 


SUMMARY 


considerable ranges. When a sharp gradient lies just 
above the projector and a layer of nearly isothermal 
water 100 ft or more in thickness lies below, ray 
theory predicts that the sound bent back up by the 
positive velocity gradient in the isothermal layer 
should be focused at shallow depths and long ranges, 
thus giving anomalously high intensities. Observa¬ 
tions made under these conditions show that the 
transmission anomaly on a shallow hydrophone is 
sometimes as much as 40 db less than normal over a 
narrow range interval several thousand yards away. 
The details of these observed effects are not in good 
agreement, however, with the exact predictions of 
ray theory. 

Deep Sound Channels 

At a depth of several thousand feet there is usually 
a deep sound channel. The effect of pressure on sound 
velocity increases the velocity at greater depths, 
and a thermocline usually present closer to the sur¬ 
face increases the velocity at shallower depths. Sound 
of frequency less than 200 cycles, for which the ab¬ 
sorption is very low, has been observed to propagate 
out for several thousand miles in such a deep channel. 
With small explosive charges, the arrivals of the 
different pulses agree with the different rays pre¬ 
dicted theoretically. The largest number of arrivals, 
with the highest observed intensity, occur just be¬ 
fore the observed sound stops entirely; these last 
arrivals are the rays coming almost straight along 
the axis of the channel. 

10.3 SHALLOW-WATER TRANSMISSION 

In shallow water, the transmission of underwater 
sound is determined primarily by the character of the 
bottom, and by the frequency of the transmitted 
sound. The state of the sea is a much more important 
factor than in deep water. Temperature gradients are 
of secondary importance. There are two situations 
in which sound conditions do not differ appreciably 
from those found in deep water: (1) soft MUD bot¬ 
tom; (2) strong positive velocity gradients (PETER 
pattern) below a directional sound source. In both 
these cases, transmission is very nearly the same as 
in deep water with the same thermal conditions. 

10.3.1 Sonic Frequencies 

Most of the information on the transmission of 
sonic sound in shallow water was obtained in harbor 


surveys. The data obtained may be summarized as 
follows. 

No systematic difference was found between dif¬ 
ferent types of bottoms, with the exception of soft 
MUD, which turned out to be a poor reflector. All 
other bottoms apparently reflect equally well. 

Transmission over sloping bottoms in the presence 
of downward refraction tends to be poor, in agree¬ 
ment with theoretical predictions. 

Over flat bottoms, at ranges greater than the water 
depth and out to several thousand yards, average 
sound transmission can be best represented by an 
inverse 1.5th power law of spreading plus an attenua¬ 
tion which appears to increase roughly linearly with 
the frequency up to about 20 kc. The transmission 
loss is, thus, given roughly by a formula 

H = 15 log r + ^(/ - 2)r + C, (10) 

where r is the range in yards, / is the frequency in 
kilocycles, and C is a constant independent of the 
range r. 

10.3.2 Twenty-four Kilocycles 

In moderately shallow water, and in the presence 
of any bottom but MUD and soft SAND-AND- 
MUD, the transmission anomaly can usually be 
represented in fairly good approximation by a 
straight line. For wind forces 0 to 2, transmission 
anomalies increase with the range at a rate of 5 db 
per thousand yards over STONY and SAND bot¬ 
toms, and at a rate of 6 db per thousand yards over 
ROCK bottoms. About half of all the runs carried 
out yield values which differ from these average 
values by no more than 2 db per kyd. 

The following special results are also worth noting. 
(1) For heavy seas, transmission is somewhat worse 
than for light seas. For wind force 3, about 1 db per 
kyd should be added to the attenuation coefficients 
given above. (2) Over sloping bottoms and in the 
presence of negative gradients, transmission is poor. 
The transmission anomaly may increase with the 
range at a rate exceeding 10 db per thousand yards. 
(3) In shallow isothermal water, transmission is at 
least as good as in deep isothermal water. (4) In very 
shallow water (5 fms deep), a series of experiments 
carried out over SAND gave very poor transmission; 
the anomaly increased at the rate of about 16 db 
per kyd. 





FUTURE RESEARCH 


241 


10.3.3 High Supersonic Frequencies 

As far as is known, transmission in shallow water 
at high supersonic frequencies is similar to that at 
24 kc, except for greatly increased absorption losses; 
transmission anomalies in the presence of negative 
gradients are linear and their slopes are somewhat 
higher than those in deep isothermal water. 

10.4 FLUCTUATION AND VARIATION 

The transmission loss measured at any instant in 
the ocean will usually differ from the value found 
several seconds earlier. This rapid change of sound 
level is called fluctuation. Measured transmission 
losses and transmission anomalies are averaged to 
smooth out this fluctuation. 

Fluctuation is invariably observed in the trans¬ 
mission of single-frequency supersonic signals trans¬ 
mitted over a path at least 100 yd long. Fluctuation is 
negligible over transmission paths of the order of 5 yd. 
Little is known concerning fluctuation over inter¬ 
mediate path lengths. For frequencies of 5 kc and 
less, fluctuation appears to be less pronounced than 
at frequencies of 10 kc and higher. The summary 
which follows is concerned only with the fluctuation 
of supersonic signals transmitted over paths at 
least 100 yd in length. 

10.4.1 Variance with Shallow Projector 

For a projector at a depth of 16 ft, the direct sound 
from an echo-ranging projector cannot be distin¬ 
guished from the surface-reflected sound. The fluc¬ 
tuation is large and inexplicably variable. Observed 
values of variance average 40 per cent with an inter¬ 
quartile spread of about 20 per cent. The variance at 
24 kc is significantly correlated with the variance at 
16 kc or 60 kc, the coefficient of correlation being 
about 0.7. 

10 . 4.2 Variance with Deep Projector 

For a deep projector and a deep hydrophone, the 
direct signal can be resolved from the surface-re¬ 
flected signal. The observed fluctuation of the direct 
signal is small; observed values of the variance at 
24 kc lie between 5 and 10 per cent and may result 
from the variability of the measuring equipment. 
The surface-reflected pulse is highly variable with a 
variance between 50 and 70 per cent. 

With explosive pulses, the direct sound can be re¬ 
solved from the surface-reflected pulse even at shal¬ 


low depths. The observed variance for the direct 
pulse is about 1 or 2 per cent if the transmission path 
lies wholly in an isothermal layer, but up to 20 per 
cent if part of the transmission path lies in the 
thermocline. 

10.4.3 Rapidity of Fluctuation 

The time during which the sound level is not likely 
to change appreciably is also variable, but seems to 
increase with increasing range. At a fixed range of 
less than a few hundred yards, the transmission loss 
for a shallow sound projector changes by about 
20 per cent on the average during 0.5 sec. At a fixed 
range of several thousand yards in the direct sound 
field, the average time for a 20 per cent change might 
be 2 sec; while in the shadow zone, this average time 
is likely to be nearer 0.02 sec. 

10.4.4 Variation 

Slow changes in the (averaged) transmission of 
sound in the sea, which take place in several minutes 
and which cannot be explained in terms of observable 
changes in the vertical temperature pattern, are 
called variations. It has been found that at 24 kc the 
variation between two transmission runs about 20 
minutes apart has an average value of about 4 db if 
only pairs of transmission runs are considered in 
which the bathythermograph pattern is significantly 
the same. This average value for the variation does 
not appear to depend significantly on range. 

10.5 FUTURE RESEARCH 

During World War II research on the transmission 
of underwater sound has been largely devoted to the 
empirical investigation of certain practical problems. 
A wealth of detailed information has been accumu¬ 
lated on the transmission loss of sound from a stand¬ 
ard echo-ranging projector under conditions likely to 
be observed in practice. Although this information 
has been useful in subsurface warfare, it has not led 
to any complete understanding of the physical proc¬ 
esses involved in underwater sound transmission. 
For example, the average attenuation in deep iso¬ 
thermal water near San Diego has been extensively 
measured, but the causes of this attenuation are 
completely unknown. 

In the years to come, research in this field will 
probably change its character. The quest for empiri¬ 
cal data on some particular situation has been carried 



242 


SUMMARY 


about as far as usefulness requires, and future studies 
will most profitably be directed to a more funda¬ 
mental investigation of the basic factors underlying 
the observed data of underwater sound transmission. 
Without such a reorientation of the basic research 
program, it will be impossible to predict the behavior 
of underwater sound under new and unexplored con¬ 
ditions. Suppose, for instance, that sound gear using 
a nondirectional supersonic projector were to be pro¬ 
posed. The transmission loss for the sound from such 
a system could not be predicted definitely from pres¬ 
ent data, which are all obtained with directional 
supersonic sources. To make such predictions would 
require some knowledge of the importance of the 
scattering of sound through small angles. Similarly, 
the attenuation of sound transmitted from a deep 
projector to a deep hydrophone cannot be predicted 
from the present empirical data taken with shallow 
projectors, but might be estimated if the basic causes 
of attenuation were known. 

In principle, the answer to any practical question 
about underwater sound transmission could be ob¬ 
tained by a program of measurements planned wholly 
for the purpose of answering that question. When 
haste is required, this is frequently the quicker 
method. When time is available, however, such 
answers can most efficiently be provided by a broad 
program designed to yield a physical understanding 
of what is happening. Such a program makes it ulti¬ 
mately possible to answer not one but a large number 
of practical questions. Thus, in the long run, im¬ 
proved technology can best be based on a foundation 
of long-term fundamental research. 

This final section gives a brief discussion of some 
of the basic physical factors that may be expected to 
be important in underwater sound transmission and 
also treats the type of observations that might be 
expected to give meaningful information on these 
different factors. 

10.5.1 Basic Factors 

The wave equation, equation (27) of Chapter 2, 
presumably governs in good approximation the prop¬ 
agation of sound waves in the interior of the ocean. 
It appears reasonable at first to investigate solutions 
of the simple wave equation, taking account of the 
presence of velocity gradients in the sea and of the 
reflections from sea surface and sea bottom. If the 
results are in flagrant disagreement with observa¬ 
tions, then the effects of the approximations entering 


into the derivation of the wave equation must be in¬ 
vestigated in detail. Apart from the validity of the 
wave equation as such, it is known that the body of 
the ocean contains scatterers (their nature uncertain) 
which deflect a fraction of the sound energy from its 
original direction of propagation. Furthermore, the 
observed absorption at supersonic frequencies far 
exceeds the value predicted on the basis of viscosity- 
alone, necessitating the assumption of additional 
dissipative processes. 

The most important problems of undemater sound 
transmission may thus be summarized under the 
following four headings. 

1. The effects of velocity gradients in the sea. 

2. Absorption and scattering in the volume of the 
sea. 

3. Surface reflection. 

4. Bottom reflection. 

Each of these topics is discussed in the following 
subsections. 

Sound Velocity 

The velocity of sound is known as a function of 
temperature, pressure, and salinity and thus can be 
calculated at any point in the ocean where these 
physical quantities are known. The refraction effects 
produced by smooth vertical changes of temperature 
have been extensively investigated theoretically, and 
the results are in general qualitative agreement with 
the observations. Since the agreement is not com¬ 
plete, however, other effects must also play an im¬ 
portant part. While the pressure is known as a func¬ 
tion of depth, changes in temperature and salinity 
over distances of a few feet have not been extensively 
measured, and the acoustic effects to be expected 
from such changes have not been thoroughly ex¬ 
plored. Microstructure of temperature and perhaps 
also of salinity may have an important effect on sound 
transmission, especially when the smoothed vertical 
gradient of sound velocity is small. Also, microstruc¬ 
ture probably accounts for some part at least of the 
observed fluctuation of transmitted sound. 

Absorption and Scattering 

The attenuation observed in deep isothermal water 
is presumably the result of absorption, that is, some 
dissipative process which converts sound energy into 
heat. Since the attenuation observed at 24 kc exceeds 
by a factor of about 100 the value predicted on the 
basis of shear viscosity alone, the principal cause of 
the observed attenuation must be some other mech- 



FUTURE RESEARCH 


243 


anism. Among the dissipative mechanisms consid¬ 
ered are compression viscosity (which, however, cer¬ 
tainly is not the principal factor at 24 kc), gas bub¬ 
bles present in the water, fish bladders, plankton, and 
thermodynamically irreversible chemical reactions, 
such as the hydrolysis of dissolved salts. 

Gas bubbles and other inhomogeneities would not 
only absorb but also scatter sound. That scatterers 
are present in the sea is known. Scattering may ac¬ 
count for part of the attenuation of highly collimated 
beams and also is probably responsible for most of 
the sound observed in predicted shadow zones in the 
presence of negative velocity gradients. 

All hypotheses concerned with the cause of the 
absorption of sound as well as with the role of volume 
scattering on sound transmission are at present 
largely speculative. Until further experimental and 
theoretical work has provided a scientific under¬ 
standing of the mechanisms involved, it will not be 
possible to predict with confidence the attenuation 
under many different conditions. 

Surface Reflection 

The change in density at the sea surface is known 
and is so large that for most practical purposes the 
density of air may be set equal to zero; that is, the 
surface is almost a perfect reflector of sound. The 
complexity of surface-reflected sound arises from the 
complicated form of the ocean surface. In principle, 
it is simply a mathematical problem to compute the 
sound reflected from any surface of known properties. 
In practice, observations are unquestionably re¬ 
quired. A thorough understanding of this topic would 
be important in studies both of fluctuation and of the 
average transmission anomaly in the surface layer. 

Bottom Reflection 

The ocean bottom may have a topography equally 
as complicated as the ocean surface. In addition, the 
relative change in the elastic parameters and in 
density across the interface is much less extreme 
than across the ocean surface, and the detailed values 
of these changes must be considered. Since the physi¬ 
cal properties of the bottom may vary with position, 
both vertically and horizontally, the problem of bot¬ 
tom-reflected sound can be very complicated physi¬ 
cally as -well as mathematically. In certain regions, 
where the bottom is flat, and of uniform composition, 
the acoustic phenomena are perhaps capable of being 
understood. Bottom-reflected sound is obviously im¬ 


portant in many situations, especially when the direct 
sound is weakened by temperature gradients. 

10.5.2 Methods 

To understand the physics of underwater sound 
transmission, each problem must be given separate 
consideration. The following methods may be ap¬ 
plicable, however, to the investigation of a con¬ 
siderable variety of problems. 

Oceanographic Measurements 

An important part of any basic research on sound 
in the sea must be the investigation of the physical 
properties of the medium in which the sound is trans¬ 
mitted. It is in terms of these properties that the 
acoustic data are presumably to be interpreted. 

In the first place, detailed measurements of the 
factors influencing sound velocity seem desirable, 
especially temperature measurements showing the 
full detail actually present in the sea. In the second 
place, detailed measurements of the shape of the 
ocean surface are required before any attempt can 
be made to explain surface-reflected sound; in par¬ 
ticular, statistical information on the spectrum of the 
surface water waves present during any interval 
seems desirable. In the third place, complete physical 
data on the ocean bottom (on topography, composi¬ 
tion, porosity and compactness, etc.) are required to 
interpret physically the data on bottom-reflected 
sound. Finally, it may be necessary to make a variety 
of physical measurements on ocean water as part of 
the attempt to identify the cause of absorption. 

Controlled Acoustic Measurements 

The experimental techniques of underwater acous¬ 
tics research will probably be developed in a number 
of directions. Greater emphasis may be expected 
on detailed accuracy of the acoustic data; probable 
errors of several decibels for a transmission anomaly 
can presumably be considerably reduced. Measure¬ 
ments involving smaller samples of the ocean may 
perhaps be anticipated with relatively complete 
oceanographic data obtained for the small samples 
investigated. Some such experiment might be devised 
for measuring the sound absorption in a relatively 
small volume. Another possible development is along 
the lines of multiple measurement, in which many 
different items are measured almost simultaneously. 
For example, the inclination of the wave front might 
be measured at the same time as its intensity with 



244 


SUMMARY 


simultaneous recordings at a number of different 
frequencies. Increasing complexity of the necessary 
equipment may probably be anticipated. 

It is possible that explosive sound may be useful 
as a research tool. As pointed out in Chapter 9, short 
explosive pulses provide resolution of the direct and 
reflected pulses even at nearly grazing angles and 
also reveal clearly any multiple ray paths that may 
be present. By means of Fourier analysis, it is possible 
also to obtain with explosive sound many of the re¬ 
sults which could be obtained by simultaneous trans¬ 
mission of many single frequencies over the entire 
spectrum. Finally, the high sound intensities possible 
with explosive pulses can provide data at longer 
ranges than are possible with standard sound pro¬ 
jectors. Thus, explosive sound would appear to be a 


valuable tool of underwater sound research, deserving 
wider application than it has had in the past. 

Regardless of what specific technique is used, the 
primary requirement for any basic experiment is that 
it be devised to give answers to certain physical 
questions rather than to operational problems. To 
satisfy this requirement, the theory underlying each 
experiment must be studied in detail before the ex¬ 
periment is actually performed to make sure that 
the results obtained will be significant. Considerable 
ingenuity may be required to find means for isolating 
the effects of the different factors involved in order 
to investigate them separately. It is only by such 
carefully designed experiments that our general un¬ 
derstanding of sound in the sea can be continually 
increased. 



PART II 


REVERBERATION 































































Chapter 11 


INTRODUCTION 


11.1 DEFINITION OF REVERBERATION 

I n any echo-ranging or listening device, the 
wanted signal is always received in the midst of a 
certain amount of extraneous noise. This background 
noise is a mixture of noises from many sources, most 
of which are operative whether the transducer is 
being used for echo ranging or for listening. Examples 
of noises which appear in both echo-ranging and lis¬ 
tening backgrounds are noise from breaking waves, 
sounds produced by such marine organisms as snap¬ 
ping shrimp, noise from the forming and collapsing 
of bubbles around the screws of the ship, noise due 
to the local motion of the water in the vicinity of 
the moving sound head, and the general din of the 
motors of the ship and auxiliaries. In both listening 
and echo ranging these noises, in varying degrees de¬ 
pending on conditions, are present to confuse the 
operator. 

Echo-ranging backgrounds, however, have another 
component all their own, which is directly due to the 
pulse put into the water. This component is called 
“reverberation,” and is the topic for discussion in 
Chapters 11 to 17. 

Reverberation is evident to the operator of echo¬ 
ranging gear as a quavering ring, which sets in as 
soon as the gear is rigged for reception, that is, as 
soon as the period of sound emission is ended. 1 This 
rolling sound has about the same pitch as the out¬ 
going pulse. Reverberation is very loud a fraction of a 
second after the ping is sent out, but diminishes 
rapidly thereafter if the receiver amplification is not 
changed. However, this rapid decrease does not neces¬ 
sarily mean that echoes from distant targets will al¬ 
ways be audible above the reverberation coming in 
at the same time. Wanted echoes also become fainter 
as the time interval between ping emission and echo 
reception increases so that echoes which would be 
audible over the remainder of the noise background 
are often masked by reverberation. 

Although reverberation in the sea shows some simi¬ 


larity to the well-known reverberations in a room, in 
many respects it is quite different. In a closed room, 
a sound is reflected with diminishing intensity back 
and forth between the walls and floor and ceiling 
until it is finally absorbed. Since the absorption of 
sound in air is relatively small, the sound energy dis¬ 
appears in appreciable amounts only at the bound¬ 
aries of the room ; and the decay of the reverberation 
is simply a measure of the decay of acoustic energy 
in the volume of air enclosed by the room. In the sea, 
there are these important differences. The sea has 
boundaries similar to a ceiling and floor (sea surface 
and bottom), but nothing like walls to interrupt the 
free passage of sound in a horizontal direction. 
Further, the sound-transmitting properties of sea 
water differ from those of air. The sea volume both 
absorbs and scatters sound energy in appreciable 
amounts. Thus, the behavior of reverberation in the 
sea is a special problem upon which little light can 
be cast by the known properties of reverberation in 
a room. When an echo-ranging pulse is sent into the 
water, some of its energy does return back to the 
transducer; but the amount of returning sound de¬ 
pends on many factors besides the rate at which 
sound energy is being removed from the volume of 
the ocean by the boundaries. 

11.2 ELEMENTARY PROPERTIES OF 
REVERBERATION 

There is every reason to believe that reverberation 
is almost always the resultant of a large number of 
very weak echoes arising from small bodies or irregu¬ 
larities in the path of the ping. These tiny targets 
may be called “scatterers.” They may be air bubbles, 
suspended solid matter, minor irregularities of the 
ocean surface and bottom, local fluctuations of water 
temperature, or any other inhomogeneities in the sea. 

Let us suppose, to begin with, that the scatterers 
producing reverberation are all identical, and are 
uniformly distributed throughout the volume of the 


247 


248 


INTRODUCTION 


ocean, and that a steady sound signal is sent out into 
this idealized medium. If we ignore, for the moment, 
the wave character of sound, and regard a traveling 
sound impulse as just a steady stream of energy, then 
each scatterer will return sound energy at a uniform 
rate, and in a receiver set up to measure the returning 
sound the reverberation will be heard as a steady 
ring of constant intensity. It is fairly obvious that in 
this situation the intensity of the received reverbera¬ 
tion should be directly proportional to the intensity 
of the outgoing signal, since an individual scatterer 
returns a fixed fraction of the sound energy incident 
on it. 

If, instead of a continuously projected signal, a very 
short pulse is put into our ideal medium, the resulting 
received reverberation will be quite different in 
character. Intuitively, we would guess that the re¬ 
verberation should gradually taper off to a small 
value in a few seconds, since by that time the pro¬ 
jected energy is already miles aw'ay from the re¬ 
ceiver. It is not difficult to estimate the rate of decay 
of reverberation in this simple situation if we ex¬ 
plicitly neglect the absorption of sound energy in the 
sea. As the sound pulse travels away from the pro¬ 
jector, it spreads out over a larger and larger volume; 
but, if we neglect absorption, the amount of energy 
included in the pulse does not change. By assuming, 
as is approximately the case, that the rate at which 
energy is scattered is simply proportional to the 
energy in the pulse, it follows that as the pulse travels 
outward from the source its energy is scattered by 
the sea volume at a constant rate. However, this 
scattered sound must make the return trip back to 
the transducer before it is heard as reverberation; and 
on this return it is weakened by inverse square 
spreading. The reverberation from a short pulse, 
then, is inversely proportional to the square of the 
range of the scatterers producing it. Since the range 
is proportional to the time, this means to the man 
with the earphones that the reverberation decays 
inversely as the square of the time. 

This result suggests that the intensity of reverbera¬ 
tion is a mathematically smooth function of the time, 
of the sort indicated by the broken line in Figure 1. 
Such a curve implies that, a short time after the cessa¬ 
tion of the projected sound, the intensity of rever¬ 
beration is great and that it fades away smoothly and 
gradually, becoming always fainter and fainter as 
time goes on. Everyone who has listened to echo 
ranging knows that this is not the case. The most 
obvious property of reverberation is its variability, 


its alternation of bursts and silences, as schematically 
illustrated by the solid line of Figure l. 2 This varia¬ 
bility is associated with the phenomenon of inter¬ 
ference. 



Figure 1. Decay of reverberation intensity. 

Interference arises because of the w r ave-like char¬ 
acter of sound. Because of interference, the reverbera¬ 
tion from n similar scatterers illuminated by the ping 
does not always have n times the intensity which 
would be observed if only one scatterer were present . 
At a particular instant, the sounds returning from 
the n scatterers may interfere destructively at the 
hydrophone so that the n sounds annul one another 
completely. Or, the n individual sounds may all com¬ 
bine constructively, so as to give n 2 times the in¬ 
tensity which would have been due to one of the 
scatterers alone. These are two extreme cases; but in 
general the n scatterers together may produce com¬ 
posite intensities ranging all the way between these 
two limits. The resultant intensity that occurs in a 
given situation depends in a critical way on the 
exact positions of the scatterers relative to one an¬ 
other. Since in the actual ocean the separations of the 
scatterers change from one portion of the ocean to 
another, it is plain that the reverberation from a ping 
should not change smoothly with time; rather, it 
should change irregularly, with bursts where the 
interference of the individual tiny echoes is primarily 
constructive, and relative silences where the inter¬ 
ference is primarily destructive. 

There is yet another complication. If the echo¬ 
ranging ship and the scatterers were all fixed in posi¬ 
tion, the pattern of bursts and silences, although com¬ 
plex, would not change from one ping to the next. 








PREVIEW 


249 


However, the echo-ranging vessel is usually rolling 
and pitching; and the scatterers in the ocean, that is, 
the air bubbles, the suspended solid matter, the tur¬ 
bulent regions, and the regions of temperature fluctu¬ 
ation are all free to move. Thus the interference pat¬ 
tern will vary widely from one ping to the next. Since 
the exact distribution of the scatterers in the ocean 
cannot be predicted, the irregularities of reverbera¬ 
tion can be described adequately only by statistical 
methods. That is, if we are to be realistic, we can 
attempt to assign values only to the average rever¬ 
beration intensity, and the average reverberation 
variability. For example, the inverse square law for 
the decay of reverberation from the volume of the 
sea, which was developed previously, could only be 
valid for the average reverberation from a series of 
pings; it would be nonsense to expect the reverbera¬ 
tion from an individual ping to decay smoothly in 
exact agreement with this law. 

In order to make clear the meaning of reverbera¬ 
tion, the scatterers in the water, such as air bubbles, 
suspended solid particles, and the like, have been 
assumed to be very nearly uniformly distributed 
throughout the sea volume. We have really been 
describing what is known as volume reverberation, 
that is, reverberation due to scatterers in the body of 
the water. However, there is also reverberation due 
to scatterers at the ocean surface and ocean bottom. 
These three types of reverberation, volume, surface, 
and bottom reverberation, behave quite differently 
on the average. For example, they set in at different 
times. Volume reverberation is evident at the mo¬ 
ment the ping is put into the water, and surface and 
bottom reverberation do not set in until the sound has 
had time to travel to these bounding surfaces and re¬ 
turn to the transducer. There are other differences as 


well, which are discussed in the main body of the 
text. 

11.3 PREVIEW 

The next six chapters summarize the reverberation 
studies carried out under the auspices of NDRC 
through the spring of 1945. Chapter 12 derives theo¬ 
retical formulas for the average reverberation in¬ 
tensity on the basis of assumptions which are ex¬ 
plicitly stated and whose validity is critically ex¬ 
amined. In that chapter, the expected intensities of 
reverberation from the volume, surface, and bottom 
are examined separately for their theoretical depend¬ 
ence on many other variables besides time; some of 
these other variables which play a major part in de¬ 
termining the reverberation intensity are the direc¬ 
tivity characteristics of the transducer, the trans¬ 
mission loss between the projector and the scatterers, 
the intensity and duration of the projected signal, 
and the scattering power of the portion of the ocean 
under consideration. Chapter 13 describes the field 
and laboratory methods which have been developed 
for the measurement of reverberation intensity and 
the analysis of the resulting data. Chapters 14 and 15 
summarize the observational information on volume, 
surface, and bottom reverberation which has been 
obtained by use of these experimental and analytical 
techniques and compare these results with the 
theoretical predictions of Chapter 12. In Chapter 16 
the variability of reverberation is examined, both 
theoretically and in the light of the observed data, 
and the frequency characteristics of reverberation 
are described. Finally, in Chapter 17 the most im¬ 
portant results presented in the body of Part II are 
summarized. 




Chapter 12 


THEORY OF REVERBERATION INTENSITY 


T he amount and character of the sound heard or 
recorded as reverberation depends not only on 
the properties of this sound in the water, but also on 
the nature of the gear in which the reverberation is 
received. The intensity of the reverberation actually 
heard or recorded, after the sound in the water has 
been converted to electrical energy by the receiver, 
amplified, and passed to the ear or recording scheme, 
will be called the “reverberation intensity,” and will 
be given the symbol G. As so defined, G equals the 
watts output across the terminals of the receiving 
gear. Although in practice the reverberation may be 
measured in terms of volts, or the height of a line on 
a motion picture film, or some other convenient 
quantity, these measurements can always be con¬ 
verted to watts output by the use of known param¬ 
eters of the receiver system. In general, the reverbera¬ 
tion intensity G is a function of time and is related to 
the sound intensity in the water by such parameters 
of the receiver system as receiver directivity and re¬ 
ceiver gain. 

Since G depends on the gear parameters, its abso¬ 
lute magnitude is usually not of great significance in 
research. For this reason it is customary to relate G 
to the reverberation intensity which would be regis¬ 
tered under certain standard conditions. This stand¬ 
ard reverberation intensity, in decibels, is called the 
“reverberation level” and will be defined precisely 
later. Reverberation levels are more useful than re¬ 
verberation intensities for comparing measurements 
made with different systems. 

This chapter is devoted to a theoretical analysis of 
expected reverberation intensities and reverberation 
levels. Formulas will be derived giving the depend¬ 
ence of these quantities on various gear parameters, 
oceanographic conditions, and elapsed time since 
emission of the signal. First, however, we must dis¬ 
cuss the scattering of sound, since scattering is 
usually regarded as the fundamental source of re¬ 
verberation. 


12.1 SCATTERING OF SOUND 

The analysis in this chapter is based on some very 
definite assumptions about the nature of reverbera¬ 
tion. It is assumed that not all of the sound in the 
outgoing ping proceeds outward in accordance with 
the elementary theory in Chapters 2 and 3, but that 
some of the sound is “scattered” in other directions. 
The reverberation is thought to be that part of this 
scattered sound which returns to the transducer. 
Volume, surface, and bottom reverberation are as¬ 
sumed to result, respectively, from scattering in the 
volume of the ocean, at the surface of the ocean, and 
at the ocean bottom. 

In an ideal unbounded fluid in which the sound 
velocity is everywhere the same, it is shown in 
Chapters 2 and 3 that sound always travels outward 
from its source along straight lines. In such a medium, 
then, scattering never occurs and no reverberation 
should be heard. There is no reason to doubt the 
validity of this theoretical result. Scattering arises 
because the ocean is not an ideal unbounded medium 
with constant sound velocity. It can be shown theo¬ 
retically that whenever a sound wave travels through 
a portion of a fluid where the density or sound velocity 
varies with position, some energy is radiated in direc¬ 
tions differing from the original direction of the wave. 
Similarly, whenever a sound wave in the ocean im¬ 
pinges on a new medium, for example a bubble, in 
which the density or sound velocity differ from their 
values in the surrounding water, energy is radiated 
in directions differing from the original wave direc¬ 
tion. 

V hether or not this deviated energy is called 
“scattered energy” is to some extent a matter of 
definition. If the inhomogeneity in density or sound 
velocity extends over a large region of space, a sound 
beam traveling through the medium may be changed 
in direction with little or no loss of energy from the 
beam; if this happens the sound wave is not regarded 


250 


SCATTERING OF SOUND 


251 


as scattered. Such changes in beam direction occur, 
for example, when the beam is refracted by a tem¬ 
perature gradient which is a function of water depth 
only. Similarly, reflection from an infinitely smooth 
plane surface sharply changes the direction of the 
beam; but since all the energy theoretically remains 
in the beam the process is not termed scattering. 

However, there are inhomogeneities of small size, 
such as air bubbles, or small irregularities in the 
ocean surface, which cause energy to be “detached” 
from the main sound beam, that is, to travel in a dif¬ 
ferent direction from that of the main beam. This 
detached energy, which differs in direction from the 
main beam and which results from local inhomogenei¬ 
ties in the ocean or bounding surfaces, is called 
scattered energy. It is apparent from this discussion 
that the distinction between scattered energy and 
nonscattered energy is not always too clear; for ex¬ 
ample, bottom reverberation received from under¬ 
water cliffs might more properly be called reflected 
energy rather than scattered energy. This possible 
confusion in nomenclature is of no immediate con¬ 
cern. The important point is that the existence of 
reverberation is predicted by theory from the known 
inhomogeneity of the ocean. 

The magnitude of the reverberation reaching the 
water near the receiver is calculable, in principle, by 
solving some differential equation which, with ap¬ 
propriate boundary conditions, takes into account 
the inhomogeneity of the ocean. Since temperature 
gradients and density gradients are small in the body 
of the sea, the differential equation would be the 
wave equation 


?2^J d j2 + ^ + d ll) 

dt 2 Vdx 2 dy 2 dz 2 /’ 


( 1 ) 


where p is the sound pressure, and c is the velocity 
of sound at the time t at the point whose coordinates 
are ( x,y,z ); this equation was derived and its applica¬ 
tion discussed in Chapters 2 and 3. The presence of 
solid particles and air bubbles, and the nature of the 
roughnesses in the ocean surface and bottom, are 
described by the boundary conditions; these condi¬ 
tions give the positions at each instant of all the 
surfaces at which the density and sound velocity 
change discontinuously and the amounts of these 
changes. 

Because of the complexity of the ocean, neither the 
function c(x,y,z,t ) nor the boundary conditions are 
precisely known. The bathythermograph gives the 
broad outlines of the temperature distribution. How¬ 


ever, the locations and magnitudes of small local 
temperature gradients cannot be determined with the 
bathythermograph, although such “thermal micro¬ 
structure” is known to exist. 1 Furthermore, the posi¬ 
tions of bubbles, solid particles, waves, etc., change 
continually and unpredictably. Even if the sound 
velocity and the boundary conditions were specified 
exactly, it would be an insuperable mathematical 
problem to solve equation (1) rigorously for p. Thus, 
theoretical formulas for reverberation cannot be de¬ 
rived by solving equation (1) with boundary condi¬ 
tions. 

Instead, we shall base our mathematical analysis 
of reverberation on several simplifying assumptions. 
Since a great deal is known about the general proper¬ 
ties of solutions of equation (1), reasonable assump¬ 
tions can be made about reverberation, even though 
a complete solution of equation (1) cannot be ob¬ 
tained. The principal assumptions which we shall 
use are: 

1. Reverberation is scattered sound. 

2. Scattering from an individual scatterer begins 
the instant sound energy begins to arrive at the 
scatterer and ceases at the instant sound energy 
ceases to arrive at the scatterer. 

3. Multiple scattering (rescattering of scattered 
sound) has a negligible effect on the intensity of the 
received reverberation. In other words, all but a 
negligible portion of reverberation is made up of 
sound which has been scattered only once. 

4. The intensity of the sound scattered backward 
from a small volume element dV is directly propor¬ 
tional to each of the three following quantities: the 
volume occupied by dV, the intensity of the incident 
sound, and a “backward scattering coefficient” desig¬ 
nated by m, which depends only on the properties of 
the ocean in the neighborhood of dV. 

5. The average reverberation intensity, which is a 
function of the time elapsed since the emission of the 
ping, is the sum of the average intensities received 
from the individual scatterers in the ocean. To ex¬ 
press this assumption in mathematical form, let 
g{t)dV represent the average intensity, t seconds 
after the emission of a ping, of the reverberation re¬ 
sulting from scattering in the volume element dV 
only. Then the average intensity of the reverbera¬ 
tion received from the entire ocean, at the time in¬ 
stant t, is given by 

Gil) = fg(t)dV (2) 

where the integral is taken over the entire ocean. It 




252 


THEORY OF REVERBERATION INTENSITY 


will be seen later that the function g{t) is zero every¬ 
where in the ocean except inside a thin, roughly 
spherical shell; this shell has the projector as its 
center, an average radius depending on the value of t, 
and a thickness depending on the length of the 
emitted signal. 

With these assumptions, it is possible to derive 
theoretical formulas for the reverberation intensity 
as a function of gear parameters and oceanographic 
conditions. This chapter will be concerned only with 
the average reverberation intensity to be expected in 
a series of pings. No attempt will be made in this 
volume to predict the level of the reverberation from 
one specified ping. The observed levels of the rever¬ 
beration from pings only a few seconds apart often 
differ by many decibels. Discussion of the average 
value of this fluctuation will be deferred until 
Chapter 16. 

Although the above assumptions can be defended, 
they are by no means obvious and require elabora¬ 
tion. In particular, it is necessary to specify carefully 
the meaning of the terms “backward scattering 
coefficient” and “average reverberation intensity,” 
which are introduced in assumptions 4 and 5. The 
average reverberation intensity is defined as the 
average from ping to ping. That is, if we measure 
the reverberation intensity on a succession of n pings 
with each measurement performed at a definite time t 
after midsignal, then the average reverberation in¬ 
tensity at time t is 

m = -£(?<(*) o) 

n i 

where (7,(0 is the reverberation intensity measured 
on the ith ping; the symbol 2 means summation over 
all the pings. 

The number of pings averaged must be large 
enough to smooth out the effects of fluctuation, yet 
not so large that such external factors as wave height, 
water depth, and amoimt of suspended matter in the 
ocean can change materially during the series of 
pings. In practice, thp number of pings averaged has 
usually been between 5 and 12, with not more than 
about 60 seconds between the first ping and the last. 
Some discussion of the validity of this averaging 
procedure is given in Chapter 16. 

Also, we must specify more exactly the meaning 
of the backward scattering coefficient m. If we con¬ 
sider a volume V made up of many small volume 
elements dV, then, strictly speaking, dV can scatter 
sound only if sound energy reaches it and if it con¬ 


tains some scattering substance. Thus, if dV lies en¬ 
tirely within some rigid scatterer, such as a bit of 
metallic dust, practically no sound reaches dV be¬ 
cause almost all the sound impinging on the scatterer 
is scattered at the surface of the scatterer. Another 
difficulty is that there is no way to predict the loca¬ 
tions of the scatterers on any one ping. For these 
reasons it is impossible to predict how much scat¬ 
tering from a specified volume element dV will occur 
on any one ping. We can, however, speak of the 
average scattering power of the ocean in the neigh¬ 
borhood of dV. The backward scattering coefficient 
m for a volume V, in the neighborhood of and in¬ 
cluding dV, is defined as follows. Let V be insonified 
by a plane wave of unit intensity n times in succes¬ 
sion. Let bi be the energy scattered per second per 
unit solid angle in the backward direction, during 
the fth trial, by the volume V. Then m for V is 
defined by 



The factor 4 t is introduced so that in cases where 
the scattering is the same in all directions, the average 
amount of energy scattered per second in all direc¬ 
tions will be just mV. With the definition of m given 
by equation (4) that the average energy scattered by 
dV per second per unit incident intensity per unit 
solid angle in the background direction is just 
(m/4w)dV, it also follows that {m/^ir)dV is just the 
intensity of the scattered sound from dV at unit 
distance from dV when the incident sound has unit 
intensity. 

Evidently the volume V in equation (4) cannot be 
chosen arbitrarily if the definition of m is to have any 
significance. V must be chosen small enough that m 
can vary with position in the ocean and can thereby 
indicate the variation with position of the average 
number and strength of the scatterers. However, 
since it is desired that m not vary discontinuously 
from point to point, V must not be chosen too small. 
Because so little is known about the scatterers re¬ 
sponsible for reverberation, it is difficult to formulate 
the conditions on V any more precisely than t his 
Some further discussion of the significance of as¬ 
sumption 4, as well as the other previous assump¬ 
tions, is given in Section 12.5. That section is not a 
complete treatment of the problems involved, but 
may assist the reader to understand the physical 
ideas underlying the derivation of the theoretical 
formulas for reverberation. 



VOLUME REVERBERATION 


253 


12.2 VOLUME REVERBERATION 

Volume reverberation is defined as sound scattered 
back to the transducer by scattering centers in the 
volume of the sea. Let a transducer be located at 
0 in deep water far from both the sea surface and sea 
bottom. This transducer sends out a pulse of sound, 
or ping, of duration r. Because the ping is of finite 
duration, a large part of the sound energy at a time 
t/2 seconds after midsignal will be contained between 
two closed surfaces Si, S 2 , portions of which are 
shown in Figure 1. If the sound velocity c in the 



<? 


Figure 1. Portion of ocean scattering sound at time 
t/2. 

ocean is everywhere the same, these surfaces are 
spheres centered at 0, with radii d/2 — ct/2 and 
d/2 + cr/2. With refraction, however, the surfaces 
may be far from spherical. The reason for saying that 
a “large part” instead of “all” the energy in the ocean 
lies in the volume between Si and S 2 is that the very 
existence of reverberation shows that some sound, 
scattered earlier out of the ping, does not he between 
these two surfaces at the time t/2. However, accord¬ 
ing to assumption 3 in Section 12.1, the amount of the 
previously scattered sound which is rescattered back 
to the receiver is negligible. Therefore, because of 
assumption 2, the only scattering taking place at 
time t/2, which is important in producing reverbera¬ 
tion, occurs at those scatterers located within the 
volume SiS 2 defined by the transmission laws of ray 
acoustics. 


Now consider the sound scattered at time t/2 by 
the volume SiS 2 . Obviously, all this sound will not 
return to the receiver at the same instant since the 
sound scattered near <Si travels a shorter distance 
than does sound scattered near S 2 . It is shown in 
Section 12.5.5 that it can be assumed as a conse¬ 
quence of Fermat’s principle, 2 that the average travel 
time of sound along the path from the transducer 
0 to any point X in SiS 2 equals the average travel 
time from X back to 0. Using this result, we can 
readily delimit the region where the sound returning 
to the transducer at the time t is scattered backward. 
If t is the signal duration, and if all times are meas¬ 
ured from the middle of the signal, the signal emis¬ 
sion starts at time —t/2 and ends at time t/2. The 
sound emitted first, at time —t/2, and returning as 
reverberation at time t, is scattered backward at 
some definite time which we shall call t'. The corre¬ 
sponding travel time 7\ out to the scatterers must be 
t' + t/ 2; the travel time back to the receiver must 
have an equal value because of our assumptions; and 
the sum of these two travel times and the time of 
emission —t/2 must equal t, the time at which the 
reverberation is received. Thus, 



and therefore 



(5) 


Similarly, the sound emitted last, at time t/2 and re¬ 
turning as reverberation at the time t, must be 
scattered at a time t" given by 


t" 



and the corresponding travel time T 2 is t" 



T 

-> or 


(6) 


Thus, all the scatterers effective in producing the 
reverberation at time t must lie between a pair of 
surfaces out to which the one-way travel times are, 
respectively, t/2 — t/4 and t/2 + t/4. These sur¬ 
faces are indicated in Figure 2 by the labels S[ and S 2 . 
If there is no refraction, these surfaces and S 2 are 
spheres with radii c(t/2 — t/4) and c(t/2 + t/4) 
respectively; the volume S[S 2 is thus a spherical shell 
with average radius d/2 and thickness cr/2. In 
general, even with refraction present, the volume 
SiS 2 is about half the volume *Si*S 2 ; that is, the 








254 


THEORY OF REVERBERATION INTENSITY 


volume in which the effective scatterers lie is about 
half the volume illuminated by the ping at time t/2. 

We shall now determine the intensity of the rever¬ 
beration which reaches 0 at time t from the volume 
element dV, located at X in Figure 2. We use the 




Figure 2. Diagrams used in developing volume rever¬ 
beration formulas. 


system of coordinate axes indicated in Figure 2; the 
origin is at 0, and the ray from 0 to X leaves 0 with 
spherical coordinates defined by the tangent to 
the ray at the origin. As drawn in Figure 2, 6 is the 
angle made by the ray OT with the horizontal plane; 
thus 9 is the complement of the polar angle made by 
OT with the vertical direction OH. The amount of 
energy which the projector radiates per second into the 
solid angle dil in the direction {9,4>) is just Fb(9,<t>)dQ,, 
where F is the emission per unit solid angle in the 
direction of maximum emission and b{9,4>) is the 
pattern function of the projector defined in Part I, 
Section 12.4.4. The sound intensity at unit distance 
(1 yd) from 0, along this ray, is therefore just Fb(9,<t>). 
If I z is the intensity at the point X, the “intensity 
diminution” between the point one yard from the 
projector and the point X may be denoted by h and 
defined by 


h = —— • (7) 

Fb(S,<t>) 

The quantity h is simply related to the (positive) 
decibel transmission loss H between the point 1 yd 
from the projector and the point X by the formula 

H = - 10 log h. (8) 

The small tube of rays emitted by the projector 
into the solid angle dfl will have, at the point X, a 
cross-sectional area, perpendicular to the sound rays, 
which may be denoted by dS. Let the volume element 
dV at X be defined as a cylindrical element whose 
base area is dS and whose height is ds, an infinitesimal 
extension along the direction of the wave propaga¬ 
tion (see Figure 2). The volume included by dV is 
therefore given by 

dV = aSds. (9) 

On the average, the sound which returns to 0 from 
X traverses the same ray traced out by the sound 
which was incident on dV; this assertion, a conse¬ 
quence of Fermat’s principle, will be defended in 
Section 12.5.5. Therefore, the scattered sound giving 
rise to reverberation has been scattered directly 
“backward.” By the definition of the backward 
scattering coefficient m, the intensity at a point, 1 yd 
from dV, of the sound which returns to 0 from dV is 
just m/4ir times the incident sound intensity at X, 
times the volume of dV, or 

—hFb(9,4>)dSds. 

■iir 

It we now define h' as the intensity diminution be¬ 
tween a point 1 yd from dV and the receiver at 0, 
the intensity of the sound reaching 0 from dV is 


47r 


hh'Fb(9,4>)dSds. 


( 10 ) 


The expression (10) gives the intensity of the sound 
scattered backward from dV in the water at the re¬ 
ceiving hydrophone. Let F' be the output of the re¬ 
ceiver in watts when a plane wave of unit intensity 
is incident on the receiver in the direction of its 
maximum response. A plane wave of unit intensity 
from some other direction ( 9,4 >) will stimulate the re¬ 
ceiver to an out put of F'b'(9,4 >) where b'{9 ,(/>) is called 
the pattern function of the receiver. Finally, a plane 
wave of intensity J incident on the hydrophone from 
the direction 0,</> will cause a watts output at the 
terminals of the receiver of 

J ■ F'b'(9,4>). 


(H) 













VOLUME REVERBERATION 


255 


With customary receivers of ordinary dimensions, 
the scattered sound returning from the ranges of 
interest (say greater than 50 yd) is for all practical 
purposes a plane wave at 0. Thus, using the results 
(10) and (11), we have 

Watts output from dV = 

^ h 2 F-F'b(0,<j>)b’(e,<l>)dSds . (12) 

47T 

In equation (12), h' has been set equal to h. This as¬ 
sumption that the transmission loss is the same on 
the outgoing and returning journeys will be justified 
on the basis of the laws of acoustics in Section 12.5.5. 
In addition, b(0,<f>) and b'(9,4>) are very nearly equal 
for most transducers. 

Using assumption 5 of Section 12.1, we next ob¬ 
tain an expression for the average reverberation in¬ 
tensity G(t) at time t. Integrating equation (12) over 
the volume between the surfaces and S 2 of Figure 2 
gives 

F F' rr 

G(t ) = -j— Jjmh 2 b(d,<t>)b'(e t <t>)dSds. (13) 

In equation (13), the dependence on range is con¬ 
tained principally in dS, h, and m; these quantities 
also depend on the direction 9,<j> at which the ray 
which reaches a particular volume element leaves the 
projector. However, equation (13) can be simplified 
as follows. To a good approximation, the extension 
of any ray between the surfaces and S 2 can be con¬ 
sidered equal to Cor/2, where Co, the average sound 
velocity along the ray, is always only slightly dif¬ 
ferent from the sound velocity at 0. Equation (13) can 
therefore be rewritten as 

G(t) = ^ ^ fmh 2 b(9,<f>)b'(0^)dS, (14) 

Z 47T J 

where the integral is to be evaluated on some average 
surface perpendicular to all the rays. It can usually 
be assumed that this representative surface is the 
surface reached at time t/2 by the sound emitted at 
midsignal; in Figure 2 this surface is labeled S 3 . 

This assumption for the surface of integration in 
equation (14) will not be valid if the average value of 
mh 2 b(0,<t>)b'(0,<l>)dS along any ray does not occur 
near S 3 . For example, this assumption fails when the 
ping length is not small compared to the range of the 
reverberation. By using the simplifying assumption 
that the sound intensity decays inversely as the 
square of the distance, it is easy to show directly 
from equation (13) that at close range (t not much 


greater than t/2), the reverberation intensity may 
not be regarded as proportional to Cqt/2; rather it is 
proportional to the factor 

Cot 1 

2" & _ ±f' <15) 

Another situation for which the average value of 
mh?bb'dS may not occur near *S 3 occurs when the 
rays are curving very sharply. For most oceano¬ 
graphic conditions, this error introduced by ray 
bending is not appreciable. However, when the layer 
effect discussed in Section 5.3 is present, the error 
might be significant. In that oceanographic situa¬ 
tion, the ping travels out of an isothermal layer into 
an underlying region of sharp temperature gradient; 
and the sound scattered from parts of S[S 2 below the 
isothermal layer has a higher transmission loss to 
the transducer than sound scattered from above the 
layer. 

Although equation (14) cannot be used as it stands 
for the calculation of volume reverberation levels, it 
nevertheless is significant. It implies that irrespective 
of the directivity pattern of the transducer, and of the 
oceanographic conditions, the average intensity of 
the received volume reverberation should be propor¬ 
tional to the ping length. This important conclusion 
is based, of course, on the various assumptions made 
previously. 

In equation (14), write dS = (dS/d$l)dQ, where dQ 
is the element of solid angle in the direction (9,<f>). 
Then equation (14) can be further simplified if it is 
assumed that the transmission loss in the ocean de¬ 
pends only on the distance traversed by the ray en¬ 
tering or leaving the transducer, and not at all on 
the direction of the ray. Then h and dS/dSl are inde¬ 
pendent of and equation (14) can be written as 

G(t) = ^ ~ Jmb(8,<t>)b'(6,<t>)dSl. (16) 

The term dS/dQ is placed in front of the integral sign 
in equation (16) because it is a measure of the trans¬ 
mission loss due to refraction; dS/dtt is, in fact, just 
the reciprocal of the intensity diminution due to 
normal inverse square divergence plus refraction, ac¬ 
cording to Chapter 3. 

Finally, if it is assumed that scattering in the ocean 
is independent of the initial ray direction ( 9,<f >), the 
backward scattering coefficient m can also be re¬ 
moved from under the integral sign. This yields as 
our end result for the average reverberation intensity 






256 


THEORY OF REVERBERATION INTENSITY 


G(t) = 


Cqt F • F'h 2 m dS 
2 4 tv dtt 


/< 


&(0,0)&'(0,0)dQ. (17) 


These latter assumptions are not always realistic. 
The assumption that transmission loss in the ocean 
depends only on the range and oceanographic condi¬ 
tions, and not at all on the initial ray direction, is 
probably a poor one for volume reverberation in a 
nondirectional transducer, because transmission loss 
in a vertical direction may differ appreciably from 
transmission loss in a horizontal direction. Even in 
a highly directional transducer of the sort used by 
the Navy for echo ranging at 24 kc, this assumption 
may be in error, since at long range rays leaving the 
projector only a few degrees apart may travel along 
widely separated paths; such a divergence occurs, for 
example, when the refraction theory predicts a split¬ 
ting of the beam. Moreover, when split beams occur, 
m will not be independent of (0,0) if the scattering 
strength of the ocean is not independent of depth. 
For, if the overall scattering strength of the sea is 
not the same at all depths, then a pair of rays which 
become widely separated by the prevailing refraction 
may reach portions of the ocean with different scat¬ 
tering strengths; in such a case m evidently will de¬ 
pend on (0,0). Of course m may always be regarded 
as an average over the entire volume of the ping, and 
thereby may be removed from under the integral 
sign in equation'(16). But if the scattering strength 
and transmission loss in the ocean really vary with 
angle within the main transducer beam, this type of 
averaging procedure makes the value of m depend on 
the directivity pattern of the gear; in this event re¬ 
moving m from under the integral sign has little 
significance. 

In equation (17) the dependence of reverberation 
on the directivity pattern of the gear is contained 
wholly in the integral, which can be evaluated from 
the known directivity patterns of the transducer as a 
projector and a receiver. If the transmission loss 
obeys the inverse square law, that is, if the losses due 
to refraction, absorption, and scattering are neg¬ 
lected, then 

, 1 dS 2 

h = - ; — = r 2 




and equation (17) becomes 
Cqt F 


G(t) = v ~2 fW,4>)b'(0,*)dO, (18) 

2 4 it r 2 J 

where r, the range of the reverberation, is equal to 
Coi/2. In this ideal case, then, the average intensity 


of the received volume reverberation varies inversely 
as the square of the time following midsignal. 

In general, however, the ocean is far from ideal and 
this simple law would not apply. To compare the 
observed time variation of the received reverberation 
in the general case with that predicted by the ideal 
formula (18), the general formula (17) is written as 


Git) 


Cqt F ■ F'm 

~2 r 2 



(19) 


or, in decibels 

10 log G(t) = 10 log 



+ 10 log (F-F') 


+ 10 log m — 20 log r + J v + 20 log ( r 2 h ) 


- 101 ogr dS- 


where 


J v = 10 log — |6(0,0)f/(0,0)dO. 
4 TV J 


( 20 ) 

( 21 ) 


The transmission anomaly A is defined (see Section 
3.4.1) by 

A = H - 20 log r. 

By comparing with equation (8), it is evident that 
A = —10 log ( r 2 h ). 

By substituting this expression for A into equa¬ 
tion (20) 

10 log G(t) = 10 log (y) + 10 log (F-F') 


+ 10 log m — 20 log r -f J„ — 2A + Ai (22) 


where —10 log r 2 (dtt/dS), the transmission anomaly 
which would result if the normal inverse square 
divergence were disturbed only by the effect of re¬ 
fraction, has been replaced by A\. It is apparent 
from the preceding discussion that the quantities 
A, Ai, and m in equation (22) must be interpreted 
as averages over that portion of the effective scat¬ 
tering volume which lies within the main transducer 
beam. 

The quantity Ai, which depends on refraction 
alone, cannot be measured directly. In principle, A x 
could be computed from the known temperature 
structure of the sea, according to the methods out¬ 
lined in Section 3.4. However, this computation is 
difficult, frequently inaccurate, and often totally im¬ 
practical because the observed bathythermograph 
[BT] pattern may not extend to a sufficiently great 






VOLUME REVERBERATION 


257 


depth. Alternatively, A x could be inferred from the 
observed transmission loss, if the losses due to at¬ 
tenuation and scattering were known. However, it is 
clear from the discussion in Chapter 5 that these 
losses are also uncertain. 

Equation (22) is the theoretical expression for the 
average intensity of received volume reverberation 
and is the one usually used in the comparison of 
theory with observation. Also, the computation of the 
scattering coefficient m from observed reverberation 
intensities is usually done by the use of this equation. 
It will be remembered that equation (22) was de¬ 
rived on the assumption that the transducer was 
infinitely far away from the ocean surface or ocean 
bottom, and therefore that sound traveled from O to 
A on only one path. In actual measurements, the 
transducer is always near the ocean surface and may 
be near the bottom as well. The presence of these 
surfaces increases the number of paths by which 
scattered energy from any point in the ocean can 
reach the transducer. Therefore, equation (22) will 
give erroneously low values for the reverberation 
intensity if alternative paths from 0 to X, of 
very nearly equal travel time, exist in the ocean. 
In the following paragraphs, we shall consider the 
error in equation (22) caused by the existence of 
such alternative paths. It should be stressed that we 
are considering here only the increase of volume rever¬ 
beration due to these additional paths. Surface or 
bottom reverberation will result from scattering when 
the sound impinges on one of the bounding surfaces, 
but we are interested here only in the reverberation 
resulting from the scattering of sound by the volume 
elements in the interior of the ocean. 

Possible combinations of alternative paths are 
pictured in Figure 3. If the ocean surface is calm, 
the case of Figure 3A, energy will reach the point 
X from 0 not only along the direct path OBX, but 
also along the path OAX as a result of specular 
(mirror-like) reflection from the ocean surface at A. 
If the ocean surface is rough, however, energy may 
reach X from 0 along a large, perhaps infinite, num¬ 
ber of paths, as indicated in Figure 3B. Because of 
the principle of reciprocity, the energy returning 
from X to 0 also travels along these additional paths. 

The existence of these extra paths tends to increase 
the reverberation intensity received at 0 at time t. 
To estimate the amount of increase, we note that for 
every possible path from 0 to X and back, there will 
exist an effective scattering volume of the type of 
SiSa in Figure 2, bounded by two closed surfaces from 


which scattered energy traveling along that path re¬ 
turns to 0 at time t. In Figure 3A, illustrating specu¬ 
lar reflection, there are four such volumes. One is 



A REFLECTION FROM MIRROR SURFACE 




C REFLECTION FROM ROUGH SURFACE 
AND ROUGH SEA BOTTOM 

Figure 3. Alternative paths from transducer to 
scatterer. 

S'iSi defined in preceding paragraphs, corresponding 
to the path OBXBO. The others correspond respec¬ 
tively to the paths OAX BO, OBX AO, and OAX AO. 
The volumes corresponding to the paths OAXBO and 
OBX AO are identical, because the travel time does 
not depend on the direction of travel along the ray; 
but the volume corresponding to OAXBO, the vol¬ 
ume corresponding to OAXAO, and the volume S[S' 2 
corresponding to OBXBO are in general all different. 
For each of these volumes there will be an integral 
similar to that of equation (13), expressing the con¬ 
tribution of the volume to G(t). Each such integral 
can be simplified to a surface integral multiplied by 
Cot/2, as in equation (14). It follows that the average 
intensity of the volume reverberation should be 
proportional to the ping length, regardless of whether 
















258 


THEORY OF REVERBERATION INTENSITY 


energy travels between the transducer and a scat- 
terer along one path, or along many paths. 

Evaluation of the various integrals of the form (14) 
corresponding to each possible route from 0 to X and 
back, is very difficult . These integrals depend on the 
reflecting power of the surface, on the depth and 
orientation of the transducer, and on how the back¬ 
ward scattering coefficient varies with the direction 
of the incident sound. Also, the possibility must be 
considered that for small values of t some of the 
integrals should not be included, since there may not 
be time for energy to reach the ocean surface along 
any path and return to 0 by the time instant t. 
High seas, with the possibility ol a great increase in 
the number of alternative paths, further complicate 
the problem. 

To avoid these difficulties, the customary pro¬ 
cedure has been to assume that equation (22) fully 
describes the volume reverberation intensity, despite 
the complications introduced by the ocean surface. 
The quantity 10 log m then becomes an adjustable 
parameter which measures not only the actual back¬ 
ward scattering power of the ocean for incident plane 
waves, but also the effective increase in the volume 
of scatterers caused by the existence of a number of 
alternative paths. 

If the water is deep and the echo-ranging gear is 
directional, the ocean surface can complicate the 
problem only if the main transducer beam strikes the 
surface. If the transducer beam is directed down¬ 
ward at a sufficiently large angle, the predictions of 
equation (22) should not be put in error by the 
presence of the surface. Using a depressed beam has 
proved to be one of the most convenient ways of 
studying volume reverberation. 

Most reverberation studies, however, have been 
made with the transducer near the surface, and the 
beam horizontal. Under those circumstances it is 
shown in Section 12.5.6 that the value of 10 log m 
computed from measured volume reverberation in¬ 
tensities and transmission anomalies by means of 
equation (22) will usually be about 3 db greater than 
the true value of 10 times the logarithm of the back¬ 
ward scattering coefficient. If the water is shallow 
enough for rays reflected from the bottom to be im¬ 
portant, no simple relation exists between the in¬ 
ferred value of 10 log m from comparison of equation 
(22) with experiment, and the actual value of the 
backward scattering coefficient. However, when the 
bottom is close enough to affect the validity of equa¬ 
tion (22), the volume reverberation will almost al¬ 


ways be masked by bottom reverberation so that the 
failure of equation (22) is of only academic interest. 

We shall next define the concepts of “reverberation 
level” and “standard reverberation level,” which 
facilitate the comparison of reverberation measure¬ 
ments performed with different gear and different 
ping lengths. From equation (13), the average value 
of the volume reverberation intensity is proportional 
to the product F F' where F is the power output 
of the projector and F’ is the receiver sensitivity. It is 
convenient to eliminate these variables in comparing 
the reverberation received on different gear. To this 
end we define the reverberation level R'(t ) as 

R'(t) = 10 log G(t) - 10 log (F-F r ). (23) 

For volume reverberation, we have specifically, from 
equation (22), 

/?'(<) = 10 log ~ + 10 log m — 20 log r + ,/„ 

-2 A + Hi. (24) 

In words, R'(t) is the level of the received reverbera¬ 
tion in decibels relative to the power output which 
would be produced at the terminals of the receiver 
by an incident plane wave, parallel to the acoustical 
axis, of intensity equal to the projected intensity on 
the axis at 1 yd. 

It is often convenient to go one step further. Since 
the intensity of reverberation is in principle propor¬ 
tional to the ping length, it is both desirable and 
practical to convert all reverberation levels to the 
same ping length. We define the standard reverbera¬ 
tion level for the reverberation at the ping length r as 
that which would have been received if the ping 
length had been some standard value r 0 . Let the 
standard reverberation level be denoted by R(t). 
Then we have 

R(t) = 10 log G(t) - 10 log (F-F') + 10 log 

(25) 

The predicted standard level of volume reverberation 
is therefore given by 

R (<) = 10 log C ~ + 10 log m — 20 log r 

+ J V - 2A + Al (26) 

The standard ping length t 0 is usually chosen as 100 
milliseconds. It is also frequently useful to convert 
reverberation levels to reverberation strengths. This 
is done by adding 40 log r to the computed reverbera¬ 
tion levels in equations (24) and (25), thereby ob- 



SURFACE REVERBERATION 


259 


taining respectively the reverberation strength or 
standard reverberation strength. 

The quantity J v , which specifies the relevant di¬ 
rectivity characteristics of the transducer, is known 
as the volume reverberation index. For standard Navy 
gear at 24 kc, J v is very nearly -25 db. It is shown 3 
that for transducers which are circular pistons J v can 
be very closely approximated by 

J v = 20 log y - 42.6, (27) 

where y is defined as in Figure 4. Numerically, y is 
half the angle in degrees in the plane 0 = 0 between 
those two rays of the composite directivity pattern 



Figure 4. Half-width y for circular piston trans¬ 
ducers. 


for which the product bb' is 0.25. Thus for a trans¬ 
ducer in which b = b',y is half the angle between the 
two rays for which the response as a projector or re¬ 
ceiver is 3 db less than the response on the transducer 
axis. The angle y is known as the “half width” of the 
composite directivity pattern bb'. Reference 3 also 
gives methods for calculating J v for transducers which 
are not circular pistons. 

12.3 SURFACE REVERBERATION 

Surface reverberation is defined as the totality of 
sound scattered back to the transducer by scattering 
centers in or near the ocean surface. This simple 
definition is not completely adequate, since it would 
make surface reverberation a particular part of vol¬ 
ume reverberation. We differentiate between these 
two types of reverberation by assuming that the sur¬ 
face reverberation arises from a thin surface layer of 
scatterers. The scatterers in this surface layer are 
assumed to owe their existence to the proximity of 
the surface and therefore differ in character from the 


volume scatterers which supposedly may be found 
anywhere in the volume of the ocean. 

The strength of these surface scatterers would be 
expected to be a function of the state of the sea sur¬ 
face, increasing with increasing agitation of the sea 
surface. In practice, surface and volume reverbera¬ 
tion are frequently distinguished from each other in 
just this way; surface reverberation is regarded as 
that part of the received reverberation which seems 
to depend on the sea state. 

We now derive an expression for the intensity of 
surface reverberation as a function of range and gear 
parameters, with the aid of Figure 5. We may pro- 



Figure 5. Coordinate system used in derivation of 
surface reverberation formula. 


ceed exactly as in the development for volume rever¬ 
beration, and arrive finally at an equation similar to 
equation (13). This equation for the surface rever¬ 
beration intensity Git) is 

F F' C 

G(t) = J mh>bmb'(0,4)dV, (28) 

where the integral is taken over that section of the 
volume Sj<S 2 which contains the surface scatterers. 
This section need not be of uniform depth, although 
it is drawn so in Figure 5. The factor m in equation 
(28) is the backward scattering coefficient in the 
surface scattering layer and is very probably a func¬ 
tion of the depth below the surface. In equation (28), 
reflection from the surface is explicitly neglected; 
that is, the ray paths are assumed to go directly from 

































260 


THEORY OF REVERBERATION INTENSITY 


0 to the volume elements at Y without hitting the 
surface. 

To evaluate the expression (28), we must first 
specify the volume element dV. It is not convenient 
to define the element dV in the same way as was done 
for volume reverberation. Instead, we set up a 
cylindrical coordinate system ( p,<t>,z ) whose axis OP 
is the vertical line through the transducer, as in 
Figure 5; and we define dV as the infinitesimal vol¬ 
ume lying between p and p + dp, <t> and <t> + d<t>, 
z and z -f dz. Then the value of dV is 


dV = p dtpdzdp. 

We integrate over the intersection of the volume 
S’xS'z with the surface scattering layer, which is as¬ 
sumed to have depth d. This volume is an annulus 
(ring-shaped figure) determined by z varying from 
zero to d, <t> varying from 0 to 2w, and p varying from 
S[ to £ 2 . Then equation (28) becomes 


Git) 


— dz I d<f> pmk-bb’dp, (29) 
47r Jo Jo Jsi 


where p is integrated from £j to £ 2 - In the ocean, it 
can be assumed that on the average sound rays are 
bent only in a vertical direction. Then, the distance 
in the p direction from £j to £2 is independent of the 
polar angle </>, but may depend on the depth z. 

In order to put equation (29) in a form suitable for 
calculations, we shall have to make several additional 
simplifying assumptions. First, we assume that the 
only factor in the expression (29) which depends on 
the depth coordinate z is the scattering coefficient m, 
and that m depends only on z. Then equation (29) 
becomes 

G{t) = / ~(f j mdz )f 0 d *£ ph-bb'dp. (30) 


This assumption is readily defended. Since the depth 
of the surface scattering layer is usually small com¬ 
pared to the horizontal range from the transducer, 
there is little difference in initial ray direction be¬ 
tween the ray which reaches a point Y' in the ocean 
surface and the ray which reaches the point Y" a 
depth d below Y' (Figure 5). Therefore, the product 
bb’ is practically independent of z in our volume of 
integration. Again, since the depth of the surface 
scattering layer is usually small, the horizontal dis¬ 
tance traversed by a ray in its passage from £j to £2 
changes but little from the top to the bottom of the 
layer. For the same reason it can be assumed that 
there is usually little difference in transmission loss 
among the various paths from the transducer to 


points in the volume of integration. 8 There is little 
reason for the scattering coefficient to vary with any¬ 
thing but depth, as long as the grazing angles of the 
rays on the surface do not vary appreciably over the 
volume of integration; this will be the case if the ping 
length is sufficiently short compared to the range of 
the reverberation. 

We may rewrite equation (30) as 


Git) 

where 



ph 2 bb'dp, 



(31) 


It should be remarked that the disappearance of d in 
equation (31) is of little consequence. There is ordi¬ 
narily no way to accurately estimate d in any par¬ 
ticular case; it is just the depth down to which scat¬ 
tered which depend on sea state appear in significant 
quantity. Without committing ourselves as to the 
exact size of d, we may give the factor m! real 
physical meaning by redefining it as 



where m is the backward scattering coefficient of the 
scatterers causing surface reverberation, is dependent 
on sea state, and is negligible below some unspecified 
depth. It seems likely that the depth at which m be¬ 
comes negligible is usually small enough so that the 
lack of dependence on 2 of h-bb' can be assumed. If 
not, or if for any other reason the assumptions used 
to derive equation (31) from equation (28) are not 
satisfied, the first integral in equation (30) cannot be 
regarded as a separate factor, and the concept of an 
overall surface scattering coefficient m! has no mean¬ 
ing. One situation in which equation (31) does not 
apply, while equation (28) does, is for surface rever¬ 
beration in the presence of sharp negative tempera¬ 
ture gradients near the range where the limiting ray 
leaves the surface. This situation is pictured in 
Figure 6. Strictly speaking, in this case the surface £2 
does not intersect the ocean surface at all; but S' 2 may 
be drawn to intersect the surface, as in Figure 6, with 
the understanding that the transmission loss is in¬ 
finite to the shaded volume. Under these circum- 


“ It must be noted that this assumption ignores the possi¬ 
bility of the image effect described in Section 5.2.1. The effect 
of image interference on the behavior of surface reverbera¬ 
tion is briefly discussed in Section 14.2.1. 




SURFACE REVERBERATION 


261 


stances, the transmission loss varies rapidly with 
depth in some portion of the volume of integration, 
and the assumptions used to derive equation (31) 
from equation (28) are not satisfied. 

We next make assumptions which enable us to 
integrate out the variable p in equation (31). The 
integral in equation (31) is taken over an annulus 
whose horizontal cross section is the ring-shaped area 
between the circles of radii PU and PV in Figure 5. 



o 


Figure 6. Region of surface scattering in presence of 
sharp downward refraction. 

If the ping length is assumed to be sufficiently short 
compared to the range, then h 2 bb' varies but little in 
the distance from U to V, and equation (31) be¬ 
comes 


r St ' / 


II 

2)^ " ( 


(pu+uvy (puy . _ 

=--- — in Fi g ure 5 

= uv(pu + = -p(UV) (35) 

where, if the ping length is sufficiently short com¬ 
pared to the range, p, the mean value of p in the 
annulus, may be assumed equal to the value of p 
where S 3 intersects the ocean surface. UV is the dis¬ 
tance on the surface from S[ to S 2 . 


P 



Figure 7. Expanded drawing of ray between projector 
and ocean surface. 


- f-f' r 2w C Si 

G(t ) = —— m' I h 2 bb'd4> I pdp (33) 

4tt Jo J Si 

F-F' r 2n r s -‘ 

= - m'h 2 I b(0,<t>)b f (d,4>)d(t> I pdp. (34) 

47t Jo Jsi’ 


The step from equation (33) to equation (34) is 
justified only if the transmission loss h is inde¬ 
pendent of the polar angle </>. With rays bending only 
in the vertical direction, there is ordinarily no reason 
why the average transmission loss should depend on 
this variable. In equation (34), 6 is some average 
angle of elevation of rays which strike the surface 
between the two circles of radii PU and PV in Figure 
5. If the ping length is sufficiently short compared 
to the range, this average value of 6 may be assumed 
to be the angle of elevation of the ray which leaves 
the projector at midsignal and hits the surface after 
a travel time t/2. In other words, this average value 
of d is the angle of elevation of that ray which passes 
through the curve of intersection of the ocean sur¬ 
face and the surface S 3 defined in Figure 2. 

Now, by simple calculus, 


Next, we evaluate p(UV). Referring to Figure 7, 
if PW = p, if ds is the increment of arc along the ray 
from 0 to IF in the time interval dt, and if dp is the 
corresponding increment of horizontal range, then 


and 


dp = ds cos a 

' t /2 


j ' t /2 /•</.2 

ds(t) COS a(t) = I C COS adt, 

0 Jo 

since cdt is always equal to ds. Also, since the bending 
is in the vertical direction only, we have by Snell’s 
law 


cos a 
c 


where c' is the velocity of sound at the surface, and 
a is the angle of elevation of the ray OW at W. It 
follows that 


- r l/2 c 2 

p = I — cos a'dt 
Jo c 


t COS a - 


2 


—— c 
c 


(36) 


where c is some average sound velocity. To calculate 


















262 


THEORY OF REVERBERATION INTENSITY 


UV, the second factor in the expression (35), we 
notice in Figure 7 that if UH lies in SJ and if the ping 
length is sufficiently short, then UWH is very nearly 
a right triangle with the right angle at //. Thus, 

uw = "" 


4 cos a 


since the perpendicular distance from to <8.3 is 
c't/4. With our assumptions, 


Thus 


UW = 


UV 


2 


UV = 


2 cos 


Substitution of this expression for UV and of ex¬ 
pression (36) for p into equation (35) gives 

c 2 t cos a c't 
2 c' 2 cos a 

(37) 

C~tr 

4 ’ 

In equation (37), c, the average sound velocity, can 
be replaced with little error in any actual situation 
by Co, the sound velocity at the transducer. Substi¬ 
tuting expression (37), modified by this replacement, 
in equation (34) gives 

- F-F' dtr r 2 * 

G(t ) = —— m'h?~- 6(0,0)6'(0,0)d0. (38) 

4x 4 Jo 

The subsequent procedure is similar to that 
adopted following equation (17); it is convenient to 
rewrite the theoretical expression (38) for reverbera¬ 
tion in terms of decibels as a function of range. As 
before we define the range r of the reverberation as 
c 0 t/ 2; this differs negligibly from the distance along 
the ray path OW of Figure 7. Proceeding as in Sec¬ 
tion 12.2, we find 

10 log G(t) = 101og(| T ) + 10 log {F-F') 

+ 10 log (y) - 30 log r + J.( 8 ) - 2 A, (39) 

where 

Js(d) = 10 log ^-f b{ 6 , 4 >)b'{e,<t>)d<i> (40) 

2-irJo 

and 2 A is the two-way transmission anomaly along 
the ray path. 

Equation (38) indicates that the intensity of sur¬ 



face reverberation, like that of volume reverberation, 
should be proportional to the ping length if the as¬ 
sumptions used to derive equation (38) are satisfied. 
If these assumptions are not valid in a particular 
situation, however, the surface reverberation inten¬ 
sity may not be proportional to the ping length. 
Frequently, these assumptions are not satisfied; thus 
the proportional dependence on ping length pre¬ 
dicted in equation (38) is not as general as the same 
dependence for volume reverberation predicted by 
equation (14). For example, if refraction is sharply 
downward, surface reverberation from ranges near 
where the limiting ray leaves the surface will not 
obey the theoretical law (38). Qualitatively, it can 
be seen from Figure 6 that at a range Cut /2 somewhat 
greater than the limiting range, a halving of the ping 
length may lead to more than a halving of the sur¬ 
face reverberation intensity, since most or all of the 
shorter ping may be too far from the surface to be 
effective in scattering. It will be recalled that this 
situation in which the proportional dependence on 
ping length predicted by equation (38) is invalid is 
just the type of situation which had to be ruled out in 
order to obtain equation (31), which in turn led to 
the result (38). 

In deriving equation (38), surface reflections were 
explicitly neglected. As explained in Section 12.2, the 
surface reflections make for alternative ray paths 
from the transducer to the scatterers. It is shown in 
Section 12.5.6 that these alternative paths usually 
cause the value of 10 log m ', computed from measured 
surface reverberation intensities and transmission 
anomalies by means of equation (39), to be about 
6 db greater than the actual value of the backward 
scattering coefficient of the surface scatterers. 

The quantity J s { 6 ) in equation (39) is called the 
surface reverberation index. In general, this index de¬ 
pends on the orientation of the projector relative to 
the vertical and on the range of the reverberation, 
and is difficult to calculate for arbitrary transducer 
orientations. When the transducer beam is nearly 
horizontal, however, the expression (40) can be evalu¬ 
ated approximately. It is shown 4 that if the trans¬ 
ducer is a large rectangular piston in an infinite 
baffle then, approximately, 


j~\{ 0 ,<t>)b'{ 6 ,<t>)d<t> = ^- —Q(0), (41) 


'0 
where 


cos 6 


(K 0 ) 


- Cm 

Jo 


ct>)b'{0,<fi)d<t>, (41a) 









SURFACE REVERBERATION 


263 


and £ is the angle of tilt of the transducer axis relative 
to the horizontal plane. The relation (41) is probably 
not valid for angles 0 greater than 30 degrees, at 
which angle the directivity pattern of any actual 
transducer is likely to differ appreciably from the 
ideal. Equation (41) was proved in reference 4 only 
for rectangular pistons. However, even for the circu¬ 
lar pistons used in most Navy gear, the use of equa¬ 
tion (41) is probably legitimate as long as the correc¬ 
tion factor 5(0 — £,0)5'(0 — £,0)(cos 0) _1 does not dif¬ 
fer too much from unity. For a horizontal trans¬ 
ducer (£ = 0), typical values of the correction 


10 log 


5(0,O)5 / (0,O) 
cos 0 


are given in Table 1. This table was computed by the 


Table 1 


6 in degrees 

i„ c b(e,o)b'(e,o) 

cos 0 

in decibels 

0 

0 

2 

-1.0 

4 

-2.0 

6 

-6.0 

8 

-12.0 


use of values of b and b' measured for the EB1-1, 
which is similar to standard 24-kc Navy gear. 5 The 
use of corrections greater than —12 db is probably 
not justified. Some further discussion of the use of 
this correction is given in Chapter 15. 

Formulas for J,( 0) = 10 log Q(0)/2ir are given in 
reference 3. For transducers which are circular or 
rectangular pistons 

JM = 10 log y - 23.8. (42) 


As was done with the expressions for volume rever¬ 
beration, we may define the surface reverberation 
level R'(t) as 

R’{t) = 10 log G{t) - 10 log (F-F') 

= 10 log (^j + 10 log ^-) ” 30 lo S r 

+ J.{9) - 2A. (43) 


Also, we define standard surface reverberation level 
R(t) as the level of the average reverberation which 
would have been received at the time t if the ping 
length had had some standard value rn instead of r. 
Then, 

R(t) = 10 log (7(0 - 10 log (F-F') + 10 log 



(44) 


so that 

R(t) = 10 log + lb 1 o ^(y) ~ 30 log r 

+ J.(6) - 2A. (45) 

Reverberation strengths can also be defined in a 
manner similar to that in Section 12.2. 

When using equations (39), (43), and (45) it is 
necessary to remember that A has been defined as 
the transmission anomaly along the actual ray path 
to the surface. This transmission anomaly may differ 
from A', the value of the transmission anomaly 
measured in the usual experimental determination of 
transmission loss. Consider specifically that the pro¬ 
jector axis is horizontal and that a ray leaving the 
projector with the angle of elevation 0 and an azimuth 
angle <£ of zero reaches the surface after covering the 
slant range r. Then from the definition of A, 

.4 = 10 log F + 10 log 5(0,0) — 10 log 7 — 20 log r, 

(46) 

where 7 is the measured intensity in db at the point 
where the ray strikes the surface. The measured 
anomaly A' is usually determined from the equation 

A' = 10 log F — 10 log 7 — 20 log r' (47) 

where r' is the horizontal range from the projector 
to the point where the ray strikes the surface. 
Neglecting the difference between r and r', we have 
from equations (46) and (47) 

A = A' + 10 log 5(0,0). (48.) 

Further, from equations (40) and (41) we have for a 
horizontal transducer 

Js(9) = 0) + 10 log 5(0,0) + 10 log b' (0,0) 

— 10 log cos 0. (49) 

Substituting equations (48) and (49) in equation (45) 
gives 

R(t) = 10 log \^J + 10 log (y) - 30 log r 

+ ./ s (0) - 2A' - 10 log 5(0,0) 

+ lOlog5'(0,O) — 10 log cos 0. (50) 

Under most circumstances cos 0 is sufficiently near 
unity and the projecting and receiving patterns are 
sufficiently symmetrical, with the result that the last 
three terms in equation (50) can be neglected. Thus, 
if the measured transmission anomaly A' defined by 
equation (47) is used in the analysis of surface 
reverberation, the correct expression for the standard 
reverberation level under most circumstances is 









264 


THEORY OF REVERBERATION INTENSITY 


R(t) = 10 log (y) + 10 log (y) - 30 log r 

+ Js(0) — 2.4'. (51) 

In other words, if A' is used instead of A in equation 
(45), then the proper surface reverberation index to 
use is J.(0) rather than J,(6). The same rule applies, 
of course, in the evaluation of equation (39) or equa¬ 
tion (43). 

12.4 BOTTOM REVERBERATION 

Bottom reverberation is defined as reverberation 
arising from sound scattered back to the tranducer 
by scattering centers in the ocean bottom. These 
scattering centers are thought to usually lie in a very 
thin layer on the bottom. Thus the formulas for the 
dependence on range of bottom reverberation will be 
similar to those formulas for surface reverberation, 
and can be derived from them by simple changes of 
notation. We have then, from formulas (39) through 


(45), 


✓ \ 


10 log 

G(t) = 10 log 

(jJ + 10 log (F ■ F') 




+ 10 log - 30 log r 

(52) 



+ Jb(0) — 2.4, 


with 

J B (0) = 10 

log f b(0,<t>)b'(6,<t>)(/(t> 

2tJo 

(53) 

R'(t) 

= 10 log G(t) 

— 10 log (F -F') 



- iui ° g Cf/ 

| + 10 log (jy) ~ 30 log r 




+ Jb{0) ~ 2.4. 

(54) 

m ) 

= 10 log G(t) 

— 10 log (F -F') + 10 log — 

T 



= 10 log (f 

') + 10 log - 30 log r 




+ Jb(Q) ~ 2.4. 

(55) 

In these formulas 

m" is the backward scattering 


coefficient of the bottom per unit area of the bottom. 
The coordinate system in equation (53) is similar to 
that in Figure 5; however, the transducer is usually 
directed downward instead of upward as in Figure 5. 
Equation (41) may be used to evaluate J b(Q) in 
equation (53); and as before J b{0) should be re¬ 
placed by J B ( 0) if the transmission anomaly as 
usually measured is used instead of the actual trans¬ 


mission anomaly along the ray path. The quantity 
m" is in general a function of the range since it 
probably depends on the angle of incidence of the ray 
on the bottom. 

It should be noted that the similarity of the 
formulas for bottom and surface reverberation does 
not imply that bottom and surface reverberation 
arise from similar mechanisms. The bottom scatter¬ 
ing originates in irregularities in bottom contour: 
these irregularities may vary from the fine separa¬ 
tions between grains of sand to such macroscopic 
irregularities as large rocks and underwater cliffs and 
valleys. 

12.5 EXPLICIT AND TACIT ASSUMPTIONS 

The preceding derivations of the theoretical for¬ 
mulas for volume, surface, and bottom reverberation 
were based on many assumptions, not all of which 
were stated explicitly. In this section we shall discuss 
briefly the significance and probable validity of these 
assumptions. Because of present uncertainties re¬ 
garding scattering sources and the infinite complexity 
of the ocean, this discussion is partly qualitative and 
not clear-cut. 

It was pointed out in Section 12.1 that though 
equation (1) governs the propagation of reverbera¬ 
tion in the ocean, a complete solution of equation (1) 
for the reverberation received under given conditions 
is not obtainable. However, certain general properties 
of solutions of equation (1) are known, and will be of 
use here. 

The strength of an acoustic disturbance can be ex¬ 
pressed either in terms of pressure amplitude or sound 
intensity. In practical applications, the sound in¬ 
tensity is a more convenient quantity; while in 
theoretical discussions based on the wave equation 
(1) the acoustic pressure is more convenient. In equa¬ 
tion (1), the sound intensity does not appear ex¬ 
plicitly; in fact, it is impossible to derive a simple 
differential equation, whose dependent variable is 
the sound intensity, which like equation (1) ex¬ 
presses the fact that the disturbance is a wave 
traveling through the ocean with the velocity c. In 
discussing the implications of equation (1), we shall, 
therefore, be directly concerned with the sound pres¬ 
sure p. To tie in the discussion with the preceding 
sections of the chapter, we must relate the sound in¬ 
tensity to the sound pressure and also to the voltage 
generated across the terminals of the receiving circuit. 

To simplify the discussion, we shall assume, to be- 





EXPLICIT AND TACIT ASSUMPTIONS 


265 


gin with, that the instantaneous voltage across the 
terminals of the receiving circuit is exactly propor¬ 
tional to the instantaneous pressure in any plane 
wave which is incident on the transducer. This as¬ 
sumption is equivalent to assuming that the re¬ 
ceiver introduces no phase shifts, or in other words, 
that it behaves like a pure resistance or like an ideal 
infinitely wide band-pass filter. Of course, actual re¬ 
ceivers never behave in this way, but it is convenient 
to postpone temporarily consideration of the effects 
caused by departure from ideal response in the re¬ 
ceiver circuit. Now suppose a plane wave of pressure 
p is incident on the transducer from a direction de¬ 
fined by the angles ( 9,<j >) of Figure 2. Then the voltage 
E across the receiver terminal is 

E = /'/S'(0,0)p, (56) 


where 0 '(6,4>) is the “pressure pattern function” of 
the receiver, and /' is the voltage across the receiver 
terminals when a plane wave of unit pressure is 
incident on the receiver in the direction of its maxi¬ 
mum response. Since there is no phase distortion, all 
the quantities in equation (56) may be assumed real. 
The rms power output resulting from E, in watts 
across the receiver terminals, will be 


E- 


Z 



(57) 


However, the scattered sound which produces re¬ 
verberation reaches the transducer from all direc¬ 
tions, and therefore cannot be regarded as a plane 
wave. For this scattered sound the pressure in the 
water at any instant is 

V = T>Pi, (60) 


where pi is the pressure in the fth plane wave which 
arrives at 0. The voltage generated across the re¬ 
ceiver terminals, by equation (60), is 


e =f'T,0\e i ,<t> l )p i (6i) 


where the angles 9 ,•,</>,• define the direction from which 
the fth plane wave reaches the transducer. The rms 
intensity resulting from equation (61) is given by 


f 2 = r 

z z L 


J2p'(0i,(t>i)pi Xj3'(0/,0,-)p/ 

i “'L j 

' J ' i i j ' 


i 9 s - j 
(62) 


where the double sum includes terms for all values 
of i and j except i equal to j. 


12.5.1 Average Reverberation 
Intensity 


where the receiver is assumed terminated in the pure 
resistance Z, and the bar indicates a time average 
over many cycles of the wave. 

Now, in a plane wave, the relation between the 
sound pressure p and the average sound intensity /, 
from Section 2.4.3, is just 

/ = ^ , (58) 

PoC 

where po is the density of water, c is the velocity of 
sound, and the bar again indicates a time average 
over many cycles of the wave. Equation (58) re¬ 
mains valid even if the plane wave is being refracted 
by velocity gradients in the ocean. From equations 
(57) and (58), we see that the rms power output 
across the terminals of our ideal receiver, caused by a 
plane wave incident on the transducer, is proportional 
to the average sound intensity in the water, further¬ 
more, by comparing equations (57) and (58) with the 
definition of F' and b'(9,<t >) in Section 12.2, it is 
evident that 

F' = , V(d,<t>) = P'K9,4> ). (59) 


One of the basic assumptions made in Section 12.1 
was that the average reverberation intensity is the 
sum of the average intensities of the individual 
scattered waves reaching the transducer. Because 
equation (62) represents the average reverberation 
intensity, while equation (57) represents the in¬ 
tensity of the individual scattered wave, it is clear 
that, this assumption will be strictly valid only if the 
double sum on the right-hand side of equation (62) 
vanishes. 

The average in equation (62) is the average over a 
large number of cycles. All the waves reaching the 
transducer have very nearly the same frequency 
when ordinary single-frequency (CW) pings are 
used. Thus, the value of the double sum on the right 
of equation (62) depends on the relative phases of the 
various waves arriving at the transducer. On any one 
ping the value of this double sum may be positive or 
negative, and its absolute value may be appreciable 
or near zero, depending on the phases. Thus, if the 
expression (62) is averaged over a number of pings, 
and if the phases vary in a random way from ping 
to ping, the double sum in equation (62) can be neg- 








266 


THEORY OF REVERBERATION INTENSITY 


lected, and we can conclude that the rms reverbera¬ 
tion intensity, averaged over a number of pings, will 
equal the sum of the average intensities received from 
the individual scatterers in the ocean. 

We may expect the phases to vary in a random way 
because of the properties of equation (1). This equa¬ 
tion implies directly that sound propagates through 
the ocean as a wave with a definite velocity, and that 
the phase of the returning wave depends on the 
travel time of the wave from the transducer to the 
scatterer and back. At 24 kc a phase shift of 27 t, 
amounting to a shift of a complete cycle, results from 
a relative displacement between two scatterers ot 
about an inch, or a difference in travel time of about 
40 nsec. Such phase changes or changes in ray 
path from ping to ping could result from thermal 
fluctuations, from the rise and fall of the transducer 
in the ocean, from wave motion, and from drift of the 
scatterers and the projecting ship. A relative dis¬ 
placement of one inch in five seconds (the approxi¬ 
mate time between pings) corresponds to a relative 
drift of only 60 ft per hr. 

It is worth stressing that the phase shifts discussed 
in the preceding paragraph are relative phase shifts 
between waves from different scattering points in the 
ocean. At any instant sound is being received from 
many different points on any one scatterer; but if the 
scatterer is a rigid sphere, for example, the phases of 
the waves arising at different points on the spherical 
surface always bear a definite relation to each other. 
These waves from the different points on the spherical 
scatterer will always combine to give the same result 
in equation (62), irrespective of relative displacement 
between the scatterer and the transducer. Thus, the 
likelihood that the double sum in equation (62) will 
average to zero over a number of pings is connected 
with what might be called the “correlation” between 
conditions at various points in the ocean. If the ocean 
were rigid, so that the relative positions and orienta¬ 
tions of the scatterers never changed, knowledge of 
the phase of a returning wave from one point in the 
ocean would completely determine the phases of re¬ 
turning waves from all other points. In this event the 
assumption that the double sum in equation (62) 
averages to zero would be more difficult to maintain. 
However, since the ocean is not rigid and the posi¬ 
tions and orientations of the scatterers change with 
time, knowledge of the phase of a wave returning 
from one point determines the phases only of those 
waves from the immediate neighborhood of the par¬ 
ticular point. 


The averaging to zero of the double sum in equa¬ 
tion (62) is made even more probable by our averag¬ 
ing procedure, which focuses attention on a definite 
instant relative to the midtime of the emitted signal. 
As the transducer drifts or otherwise changes its posi¬ 
tion in the ocean, the reverberation received at a 
definite instant comes from different points of space 
on different pings. In many cases, the phases of the 
scattered waves returning from these two portions ot 
space will be almost completely uncorrelated; in such 
cases, random phase relations on successive pings are 
even more likely. 

In the absence of definite knowledge about the 
scatterers responsible for reverberation, it is not 
possible to make this argument about equation (62) 
more precise. However, it is apparent from this dis¬ 
cussion that in all types of reverberation there are a 
number of mechanisms that can cause random varia¬ 
tions in the phases of the individual returning scat¬ 
tered waves. When these phases are truly random 
the assumption involved in averaging the individual 
scattered intensities, to get the average reverberation 
intensity, is justified. 

12.5.2 Definition of Backward 

Scattering Coefficient 

The backward scattering coefficient was defined for 
a volume small enough so that its relevant properties 
do not change too sharply with changes of position 
inside the volume, and large enough that it contains 
a reasonable number of scatterers. After an explicit 
assumption that the scattering by such a volume is 
proportional to the volume, the scattering coefficient 
of V was defined by formula (4). 

It is easy to see that this assumption should be 
valid if the double sum in equation (62) vanishes. If 
this term vanishes, the total reverberation intensity 
will be just the sum of the average intensities of the 
waves from the individual scatterers; therefore, it 
should be proportional, on the average, to the size 
of the scattering volume. 

12.5.3 Duration of Scattering by a 

Scatterer 

In Section 12.1, it was assumed that scattering 
from an individual scatterer begins the instant sound 
energy begins to arrive at the scatterer and ceases the 
instant sound energy ceases to arrive. 

In considering this assumption, we must recognize 




EXPLICIT AND TACIT ASSUMPTIONS 


267 


that the discussion to this point has glossed over the 
fact that neither the outgoing ping nor the scattered 
sound which reaches the transducer is really single¬ 
frequency sound. No sound of finite duration can be a 
pure single-frequency sound since the latter theo¬ 
retically lasts an infinite time; it can be shown that 
the relation between the time duration 8 t of a ping 
and the width 8 f of the frequency band making up 
the pulse is approximately 

Sfdt = 1. (63) 

In general, the relation between any time-dependent 
signal F(t) and the frequency spectrum of the signal 
is given by Fourier’s integral theorem. 6 That is, the 
signal can be written in the form 



where A (w) is determined by the equation 



Equations (64) and (65) are just generalizations of 
corresponding equations applicable to Fourier series 
of periodic functions. In these equations F(t) and 
A (oj) are generally complex, and o> can be interpreted 
as equal to 2 irf where/is the frequency of the spectral 
component corresponding to a>. 

It is possible, therefore, to make a frequency 
analysis of any given ping, using equation (65). This 
frequency analysis can then be used to obtain a formal 
solution of equation (1). For, because of the linearity 
of equation (1), if the scattered sound reaching the 
receiver as a result of emission of the continuous 
sound e iut is B(u)e iat , then the pressure of the scat¬ 
tered sound reaching the transducer as a result of 
any given ping is 

p(f) = -4= f ACu)B(a)^edu, (66) 

V2 W-°° 

where A (a>) is given by equation (65) in terms of the 
pressure variation F(t) of the outgoing ping. It is 
necessary to qualify equation (66). Because the 
boundary conditions and the velocity of sound at any 
point are changing with time, the scattered sound 
which reaches the transducer is not a pure sound, 
even though a pure sound e lb>t is emitted. Thus al¬ 
though the pressure of the scattered wave can always 
be presented as a Fourier integral of the form (64), 
equation (66) is not rigorously true, if B(a>) and A(ui) 
are defined as above. However, for the purpose of 
investigating the validity of the assumption under 


consideration, these effects of time variation can be 
neglected, and equation (66) accepted as valid. In 
addition, for simplicity, the velocity of sound c in 
equation (1) can be assumed constant. 

By using equation (66), the dependence on time of 
the pressure p(t) of the scattered sound reaching the 
transducer can be calculated for pings of any length 
and for various types of scatterers. If p(t) is plotted 
as a function of the time following the emission of the 
ping, it turns out that p(t) is always zero until the 
sound has had time to travel to and from the nearest 
point on the scatterer. In other words, scattering 
from an individual scatterer actually does begin at 
the instant that sound energy begins to arrive at the 
scatterer. It is not possible to show in general that 
scattering ceases at the instant the sound energy 
ceases to arrive at the scatterer. However, for the 
special case of an infinitely rigid spherical scatterer, 
it is possible to show that the duration of the scat¬ 
tered sound received from the sphere is the same as 
the duration r of the outgoing ping, as long as the 
relation 

- « r (67) 

c 

is satisfied, where D is the diameter of the sphere. The 
significance of equation (67) is simple; it means that 
the scattered sound will have the same duration as 
the initial ping, provided that the travel time of the 
sound across the sphere is negligibly short compared 
to the duration of the initial ping. This result is not 
unexpected. 

The reason why no general proof can be given for 
the validity of the second part of the assumption 
under discussion, namely, that scattering ceases at 
the instant sound energy ceases to arrive at the 
scatterer, is easily understood. Any real not infinitely 
rigid scatterer, such as a bubble, will have definite 
resonant frequencies which will be excited by the 
incident sound, and the scatterer may continue to 
radiate sound at its resonant frequencies long after 
the ping has passed by. Also some sound may enter 
the scatterer and be reflected back and forth inside 
the scatterer a number of times before it is scattered 
back toward the transducer. If the scatterer is large 
and many such reflections are possible, the duration 
of the scattered sound will be longer than r. Despite 
these difficulties it can be argued that the assumption 
can be regarded as valid. There is good reason to 
doubt that resonant bubbles or other resonant scat¬ 
terers play a large part in reverberation; in any case, 







268 


THEORY OF REVERBERATION INTENSITY 


the reradiated sound from such seatterers would die 
out in a very short time compared to the duration 
of even a 1-msec ping. For most seatterers equation 
(67) will be satisfied for pings of ordinary length al¬ 
though it may sometimes not be satisfied with scat¬ 
tered such as rocks on the bottom, for 1-msec pings. 

In the light of the discussion in Section 12.5.1, the 
distance D in equation (67) must be interpreted as 
the diameter of the volume within which there is ap¬ 
preciable correlation between the phases of waves re¬ 
flected from different points in the ocean. Scattering 
volumes separated by so large a distance that there is 
little correlation may be considered unrelated scat¬ 
tered. In the absence of definite knowledge about the 
seatterers, it is difficult to make the argument pre¬ 
cise, but it seems unlikely that there will be apprecia¬ 
ble correlation over a distance as great as a yard, 
which is about the length of 1-msec ping. 

Along with assumption 2, Section 12.1, it has been 
tacitly assumed, in the derivation of the theoretical 
reverberation formulas that the outgoing ping is 
square-topped (that is, that the intensity rises 
abruptly to its steady-state value at the beginning 
of the ping, remains constant until the end of the 
ping, and then drops suddenly to zero), and that the 
wave received from any scatterer reproduces the 
shape of the outgoing ping. It is possible to make the 
outgoing ping very nearly square-topped, but it is 
apparent from equation (66) that the shape of the 
waves returning from each scatterer is not necessarily 
the same as the shape of the outgoing ping. However, 
if the ping does not include too wide a frequency band 
(that is, is not too short) and if the scattering co¬ 
efficients of the various types of scattered do not 
vary too rapidly with frequency, then in equation 
(66), if sound is being received from only one scat¬ 
terer, B(u ) is nearly independent of frequency, and 
the returning scattered wave does very nearly repro¬ 
duce the wave form of the outgoing ping. It will be 
seen in Chapters 4 and 5 that even in a 1,000-c band, 
scattering coefficients in the ocean apparently change 
very little, so that, from equation (63), square-topped 
1-msec pings should result in square-topped scattered 
waves. 

W e can now see the significance of the assumption, 
made at the beginning of Section 12.5, that the in¬ 
stantaneous voltage induced in the receiving circuit 
is exactly proportional to the instantaneous pressure 
of the sound arriving at the transducer. If this is not 
the case, that is, if there is phase distortion or ampli¬ 
tude distortion in the receiver, then the reverbera¬ 


tion resulting from any one scatterer will not have 
the square-topped shape of the outgoing ping, and 
the formulas which have been derived will be in error. 
Thus, if measured reverberation intensities are to be 
comparable to the theoretical formulas of Sections 
12.2 to 12.4, it is necessary to use flat wide-bancl 
systems which have little transient response to the 
sudden changes of reverberation intensity. The use 
of narrow-band systems with high transients will 
usually cause the reverberation received from any 
scatterer to last longer than the outgoing ping and 
have a shape different from that of the ping. Since 
this distortion will decrease with increasing ping 
length r, deviation will result from the predicted 
proportionality of R'(t) on ping length r in equations 
(24), (43), and (54). In order to derive appropriate 
theoretical formulas for such systems it would be 
necessary to write, using equation (66), 

E(t ) = f A(u>)B(u,)C(co)e iut da> 

V 27T J - ® 

for the voltage induced across the receiver terminals 
by each scattered wave where C(«) describes the 
frequency response of the gear. The expression for the 
average reverberation intensity would then involve 
an integration over the frequency band included in 
the ping and passed by the equipment. It may be re¬ 
marked that there is an inherent dependence of re¬ 
sponse on frequency in any directional transducer, 
because the pattern functions b(6,<p) and b'{0,<t>) are 
always functions of frequency. Thus, it may be neces¬ 
sary to consider further the effect of directivity on the 
theoretical reverberation formulas for situations in¬ 
volving very short pings and highly directional 
transducers. 

12.5.4 Neglect of Multiple Scattering 

We have assumed that the sound reaching the 
transducer as reverberation has been scattered only 
once. It is easy to see that the validity of this assump¬ 
tion depends on the range of the received reverbera¬ 
tion. For, as the ping proceeds out from the trans¬ 
ducer, it loses more and more energy by scattering; 
the scattered waves are of course no different from 
any other sound waves and are themselves scattered. 
Eventually, therefore, a range is reached at which the 
ratio between the singly scattered sound returning 
to the transducer and the multiply scattered sound 
returning is no longer large. The value of this range 
depends on the amount of scattering which takes 




EXPLICIT AND TACIT ASSUMPTIONS 


209 


place in the ocean. Thus, the validity of this assump¬ 
tion, like that of all the other assumptions we have 
made, depends on the properties of the scatterers in 
the ocean. 

In considering the validity of this assumption, we 
may confine our attention to multiple scattering in 
the body of the ocean, since there is little likelihood 
of direct multiple scattering from one surface scat- 
terer or bottom scatterer to another. Experiments on 
volume reverberation 7 have shown that at short 
ranges (up to a few hundred yards) multiple scat¬ 
tering in the body of the ocean probably can be neg¬ 
lected. If scattering in the body of the ocean is 
isotropic, that is, if the backward volume-scattering 
coefficient m is really the average amount of energy 
scattered in all directions per unit intensity per unit 
volume, then it can be concluded from the magnitude 
of m that multiple scattering is certainly negligible 
at all ranges of interest in echo ranging. 

However, it is possible that forward scattering in 
the ocean is appreciably greater than backward 
scattering. It has been suggested that the high at¬ 
tenuation of sound in the ocean at supersonic fre- 
quencies results from forward scattering of sound by 
the temperature microstructure in the ocean. On 
present evidence, it seems unlikely that forward 
scattering alone can account for attenuation in the 
ocean, b but if appreciable wide-angle scattering does 
occur, then at long ranges the neglect of multiple 
scattering in the theoretical reverberation formulas 
is not justified. The predicted dependence of volume 
reverberation on range [equation (24)1 would be 
changed if volume reverberation contained much 
multiply scattered sound. The evidence discussed in 
Chapter 14 suggests that multiple scattering in the 
ocean probably can be neglected at ranges of opera¬ 
tional interest in echo ranging. However, more evi¬ 
dence is needed before any definite conclusions can 
be reached. 

12.5.5 Fermat’s Principle and the 
Principle of Reciprocity 

It was important to show that multiple scattering 
can be neglected in the computation of reverberation 
intensity since that assumption enabled us to deline¬ 
ate the volume »SVS 2 in Figure 1 within which ap¬ 
preciable scattering is taking place. The determina¬ 
tion of volume S'iS 2 (Figure 2) was then based on an 
application of Fermat’s principle. Fermat’s principle 


is a theorem about the properties of equation (1); 
it states that when a sound travels between two given 
points, it always follows a path such that its travel 
time is a maximum or a minimum. 2 This maximum 
or minimum value is the same no matter which of the 
two points is the starting point and which the finish¬ 
ing point. Thus, provided the refraction conditions 
and boundary conditions are not changing with time, 
the ray paths and travel times from the transducer 
out to a scatterer, and from the scatterer back to the 
transducer, are exactly the same. However, refraction 
and boundary conditions in the ocean are not con¬ 
stant, with time. The existence of thermal fluctua¬ 
tions, and the fact that BT patterns vary from one 
hour to the next, show that refraction conditions 
change with time; and surface waves are an example 
of changing boundary conditions. 

Complete elucidation of the effect of these chang¬ 
ing refraction and boundary conditions would be 
highly complicated, and, as usual, lack of information 
about the scatterers would make it difficult to be 
precise. However, it seems justifiable to assume that 
short-term fluctuations in thermal microstructure, 
or such variations in boundary conditions as waves 
or random movements of the scatterers, do not 
modify the average equality of the travel times and 
ray paths. Moreover, the long-term variations evi¬ 
dent on the BT trace are too slow to affect the aver¬ 
age equality in a series of pings lasting about a 
minute. Thus, it appears that the use of Fermat’s 
principle was justifiable in Section 12.2, in the 
delineation of the effective scattering volume S[S' 2 . 

Another theorem about the properties of equation 
(1) is the principle of reciprocity. In the ocean, trans¬ 
mission loss is thought to be a combination of ordi¬ 
nary inverse square spreading, refraction, absorp¬ 
tion, and scattering. According to the principle of rec¬ 
iprocity, 3 if the refraction and boundary conditions 
are not changing with time, then that part of the 
transmission loss which is due to inverse square 
spreading and refraction will be the same for trans¬ 
mission from a nondirectional projector at 0 to the 
point X as for transmission from a nondirectional 
projector at X to 0. Of course, the source at 0 (the 
echo-ranging projector) is not nondirectional; and 
the source at X (the scatterer) may not be nondirec¬ 
tional, since scattering is not necessarily the same in 
all directions. For directional sound sources, the 
reciprocity theorem requires modification, because of 
the possibility of reflections and multiple paths be¬ 
tween 0 and X. However, along any definite ray path 


This point is discussed in Section 5.4.1. 




270 


THEORY OF REVERBERATION INTENSITY 


the principle still holds that the transmission losses 
due to inverse square spreading and refraction from 
0 to X and X to 0 are the same along that ray, ir¬ 
respective of the directivities of the sources. In addi¬ 
tion, the principle of reciprocity applies also to ab¬ 
sorption losses 9 if absorption in the ocean arises from 
so-called linear processes. Actually, studies of trans¬ 
mission loss show that the processes involved in the 
transmission of sound in the ocean are only imper¬ 
fectly understood; but there appears to be no justi¬ 
fication at this time for ascribing the absorption 
losses of sound waves of ordinary amplitudes to non¬ 
linear processes. 

Because the scattering coefficients are so small, 
transmission losses due to scattering may be neglected 
at all ranges of interest in echo ranging. It follows 
therefore from the preceding paragraph, and from the 
definitions of the quantities h and h' in Section 12.2 as 
transmission losses along the ray, that the principle 
of reciprocity may be applied to transmission be¬ 
tween the points 0 and X of Figure 2 if refraction and 
boundary conditions are constant . It is still necessary 
to consider the effects of the variation of these con¬ 
ditions with time; but by an argument similar to that 
made in the discussion of Fermat’s principle, it seems 
valid to assume that these time variations will not 
affect the relation between the transmission losses of 
the outgoing and incoming sound on a series of pings 
lasting about a minute. In other words it appears 
justifiable at this time, in the light of the principle 
of reciprocity and our present understanding of 
absorption losses, to assume that on the average the 
one-way transmission losses h and h' are equal. 

12.5.6 Effect of Surface Reflections 

The complications induced by such variations in 
boundary conditions as rough seas are exemplified by 
the difficulties encountered in extending equation 
(20), derived for an infinite unbounded ocean, to the 
more nearly realistic semi-infinite ocean. In rough 
seas (Figure 3B), with the transducer horizontal, it is 
very difficult to determine exactly the effective vol¬ 
umes from which reverberation is being received at 
any instant. In calm seas (Figure 3A), however, the 
volumes corresponding to the various alternative 
paths should be very nearly identical at ranges 
greater than a few hundred yards, since at these 
rather long ranges the path differences between OAX 
and OBX are usually very small. These volumes will 
all be about half of the original volume Sj&j because 


of the presence of the ocean surface; at long ranges 
and shallow transducer depths the initial angle of ray 
elevation 6 is restricted almost completely to values 
between — x/2 and 0 in the integral (14), instead of 
varying from — ir/2 to r/2, as it does when the ocean 
surface is far away. Since all four of the scattering 
volumes described in Section 12.2 are very nearly 
equal, all of the integrals of the form (13) correspond¬ 
ing to these volumes should also be nearly equal, be¬ 
cause at these ranges the rays OAX and OBX (Figure 
3A) leave the transducer in almost the same direction, 
and because in a calm sea the reflection coefficient of 
the surface is very nearly unity. Thus, in a calm sea, 
with the transducer near the surface and the beam 
horizontal, the total intensity of the received volume 
reverberation is obtained by adding up four integrals 
of the form (13), with the region of integration for 
each just half the volume 

Therefore, the presence of the surface increases the 
received volume reverberation under these conditions 
to about double the value predicted by equation (17), 
or to a value about 3 db greater than predicted by 
equation (22), with the important proviso that the 
quantities A and Ai used in that equation are the 
transmission anomalies that would have been meas¬ 
ured if there had been no reflected rays. The trans¬ 
mission anomaly is usually obtained by measuring 
the transmission loss from a point about 100 yd from 
the transducer. If so, it is easily seen that with shallow 
transducers the inferred transmission anomaly is 
about the same as the transmission anomaly that 
would have been measured if the surface was far 
away. It follows, therefore, that in calm seas, with 
shallow transducers and horizontal beams, the value 
of 10 log to computed from measured volume rever¬ 
beration intensities and transmission anomalies by 
means of equation (22) will be about 3 db greater 
than the true value of 10 times the logarithm of the 
backward scattering coefficient . 

For rough seas it is not possible to make so precise 
an analysis. However, it can be argued that the dif¬ 
ference between the computed and actual value of 
10 log to will be about 3 db in rough seas also since on 
the average the existence of many paths (Figure 3B) 
will be compensated for by the loss of reflecting 
power of the surface. For surface reverberation the 
existence of surface-reflected paths causes the value 
of 10 log to' computed from measured surface rever¬ 
beration intensities by means of equation (39) to be 
about 6 db greater than the actual value of the back¬ 
ward scattering coefficient of the sui'face scatterers. 



EXPLICIT AND TACIT ASSUMPTIONS 


271 


The reason for the 6-db value for the case of surface 
reverberation is that in equation (28) the volume of 
integration explicitly includes only the semi-infinite 
region below the ocean surface. Thus in calm seas the 
total intensity of the received surface reverberation 
is obtained by adding up four integrals of the form 
(28) without any reduction in the volume of integra¬ 
tion. However, this conclusion depends on the 
properties of the surface scattering layer. If the sur¬ 
face scattering layer absorbs sound very strongly, 
sound may never be able to penetrate the layer to 
reach the actual air-water interface. In this event, the 
inferred value of m' from equation (39) will equal the 
true value of the backward scattering coefficient of 
the surface scatterers. A situation of this sort in which 
the surface layer is assumed to consist of a dense layer 
of resonant bubbles is discussed in Section 14.2.5. 

If the water is shallow enough for rays reflected at 
the bottom to be important, the situation becomes 
more obscure; as shown in Figure 3C, some of the 
possible paths between 0 and X involve both re¬ 
flection at the sea surface and reflection at the sea 
bottom. It has already been pointed out that in this 
situation no simple relation exists between the in¬ 
ferred value of 10 log m and the true value of the 
backward scattering coefficient. In fact, it may be 
remarked generally that equations (20), (39), and 


(52), for reverberation from the volume, surface, and 
bottom, respectively, are invalid when ray paths in¬ 
volving several reflections between the projector and 
the scatterer become important. 

12.5.7 Overall Evaluation 

This section has been concerned with the physical 
ideas behind the assumptions which have been used 
to derive the theoretical formulas for reverberation. 
It has been seen that most of the assumptions used 
are probably justified, but that no definite proof of 
their validity is possible at this time. If the assump¬ 
tions are not satisfied, reverberation may not depend 
on range and ping length in the manner predicted by 
the theoretical formulas (24), (43), and (54). None of 
the considerations of this section affect the possibility 
of using these formulas as an empirical means of in¬ 
vestigating reverberation and computing in each 
case a value of the backward scattering coefficient 
from comparison of the theoretical formulas with 
experiment. However, if the assumptions which have 
been made are not justified, the magnitude of the 
backward scattering coefficient deduced in this way 
will not have the simple physical significance implied 
in assumption 4 of Section 12.1. 




Chapter 13 


EXPERIMENTAL PROCEDURES 


T his chapter describes the principal methods 
which have been used in the gathering and 
analyzing of reverberation intensities. It will be seen 
that the techniques for the study of reverberation 
have been greatly improved since the first studies. A 
great many systems have been conceived for such 
studies; but only those which have actually been put 
into operation and used extensively in the gathering 
and processing of data will be discussed here. Refer¬ 
ences to sources which give more detailed informa¬ 
tion are included in the body of the chapter. 

The experimental determination of the frequency 
characteristics of reverberation will be discussed in 
Chapter 16. 

13.1 EQUIPMENT AND FIELD 
PROCEDURES 

Reverberation measurements have been made 
under a wide variety of oceanographic conditions, 
over many different types of bottoms, and at water 
depths between 10 and 2,500 fathoms. The most 
common projector depth has been 16 ft, but oc¬ 
casionally various other projector depths have been 
used. Most of these reverberation studies have been 
made by UC’DWR; quite recently, however, WHO! 
has undertaken a reverberation program of its own. 
Although differing in details, the field procedures 
have in all cases been similar in broad outline. 

In the early measurements off San Diego, made 
aboard the USS Jasper (PYcl3), this procedure was 
followed. Upon arrival at the chosen location, the 
main engines of the vessel were shut down, and the 
Jasper was permitted to drift freely. The rate of 
drift during the working clay varied between 0.5 
knot and 2 knots, depending on the wind velocity 
and ocean currents. The transducer units with their 
supporting frames were hoisted by means of an 
electrically driven winch and boom, given the de¬ 
sired orientations, swung over the rail, and lowered 
into the water to the working depth. With the sound 


gear overside, the projector and hydrophone cables 
were led through a doorway into the wardroom, and 
attached to the respective pieces of equipment. 

In later studies, the depth to which the transducer 
was to be lowered and the angle the transducer was 
to make with the vertical were “built in” to the 
equipment at the time of installation. Hoist train 
mechanisms were provided, which lowered the trans¬ 
ducer to the working depth from sea chests recessed 
in the keel. The bearing of the transducer in the 
horizontal plane was adjusted by means of a remote 
control training system. In these later modifications, 
the transducers were permanently wired to terminal 
boards, from which they could be connected to the 
regular electronic equipment or “sound stack,” or to 
specially constructed research stacks. 

After all connections are made, a sound pulse of 
controllable duration is sent into the water by means 
of a keying arrangement. As a result of this pulse, 
scattered sound returns to the transducer and gener¬ 
ates a voltage in the receiving circuit. This voltage, 
after amplification, is recorded as reverberation. 
Somewhat different methods are used by UCDWIt 
and WHOI for recording the reverberation. Systems 
for measuring reverberation intensities are discussed 
in detail below. 

13.1.1 Transducers and Electronic 
Equipment 

Most reverberation measurements have been taken 
at a frequency of 24 kc, which is the prescribed 
frequency for most Navy echo-ranging gear. Several 
types of transducer units have been employed at San 
Diego. Most of the early data 1 were obtained with a 
pair of similar magnetostrictive units (QCH-3), one 
used as a projector and the other as a receiver; some 
ot the data reported there were obtained with a 
crystal transducer, the QB. The QB crystal unit 
proved to be superior to the QCH-3 units for rever¬ 
beration studies because of its higher power output 


272 


EQUIPMENT AND FIELD PROCEDURES 


273 



Figure 1 . Schematic arrangement of apparatus employing QCH-3 units (equipment A of text). 


when projecting sound and its better response when 
receiving. Most of the more recent reverberation 
studies off San Diego have been performed with 
crystal transducers. 

The transducer alone is not capable of sending out 
pulses and detecting incoming sounds. There must 
also be equipment which delivers electrical energy to 
the projector and amplifies and modifies the small 
electrical impulses at the terminals of the receiver, 
thereby converting them into a detectable form. In 
the projector circuit of this auxiliary electric equip¬ 
ment there is an oscillator which generates an elec¬ 
trical signal of the desired frequency, and a power 
amplifier. In the receiver circuit, there is usually a 
preamplifier which takes the output at the terminals 
of the hydrophone and amplifies it somewhat, and 
then another amplifier, whose output is connected to 
the recording mechanism. Somewhat different re¬ 
cording techniques have been used by UCDWR and 
WHOI. At UCDWR, the voltage developed by the 
returning reverberation is usually fed into a cathode- 
ray oscillograph so that the instantaneous deflection 
on the cathode-ray screen is proportional to the 
instantaneous voltage developed in the receiver. The 
cathode-ray deflection as a function of time is re¬ 
corded in permanent form by the use of a camera 
with continuously moving film. In the technique 
used until very recently at WHOI, the current gener¬ 
ated in the gear by the reverberation activated a 
galvanometer, which in turn threw a light beam on 


a moving roll of sensitized paper. The newest WHOI 
equipment uses a cathode-ray oscillograph and a 
camera, but is different from UCDWR equipment in 
a number of other features. Usually inserted some¬ 
where in the receiving circuit is heterodyning equip¬ 
ment, which converts the incoming high-frequency 
energy into energy within the range of audible fre¬ 
quencies and thus permits listening to the returning 
reverberation by ear. This heterodyned signal may 
be recorded, if desired. 

In the following paragraphs, we shall discuss in 
more detail the principal electronic setups which 
have been used in making reverberation measure¬ 
ments. For convenience, these setups will be identified 
by the letters A, B, C, D, E. Setups A and B were 
used at UCDWR prior to January 1943; in later 
UCDWR studies, setup C was used aboard the 
Jasper and setup I) abroad the Scripps. Setup E has 
been used at WHOI. 

Equipment A 

This equipment 2 employed a pair of QCH-3 
transducers, one used as a projector and the other as 
a receiver. Driven at 23.45 kc, the QCH-3 projector 
generated a sound pressure on the axis of 88.5 db 
above 1 dyne per sq cm at 1 yd with a total acoustic 
power output of 1.4 watts. A block diagram for this 
system is given in Figure 1. The ping was started and 
completed by closing and opening the ground circuit 
in the oscillator-driver stage by use of an electronic 



















































274 


EXPERIMENTAL PROCEDURES 


keying relay. The ultimate keying control was a 
synchronous motor-driven pair of shafts. Disks 
affixed to these shafts operated microswitches which, 
in turn, controlled the circuit containing the keying 
relay. By adjusting these disks, it was possible to 
choose any ping length between zero and several 
hundred milliseconds, and to control the interval 
between pings. 

Because reverberation invariably decreases rapidly 
with time, the receiving system must be specially de¬ 
signed to handle a wide voltage range. For this pur¬ 
pose, a variable resistor was built into the receiving 
preamplifier, the amount of gain being controlled bv 
relays. By a keying arrangement similar to that for 
controlling the ping length, the equipment covdd be 
adjusted so that a predetermined amount of resist¬ 
ance could be removed from the receiving circuit at 
any desired time after midsignal. Thus, as the rever¬ 
beration intensity decreased, the gain of the pre¬ 
amplifier was increased in steps. 

The output voltage from the receiving amplifiers 
was fed directly to the plates of the horizontal deflec¬ 
tion circuit in a Du Mont Model 175A oscilloscope 
using a short-persistence screen; the vertical deflec¬ 
tion circuit was not connected. A continuous record 
of the oscilloscope deflections was obtained by the 
use of a fixed optical system and a camera with 
moving film. Since the spot on the screen moved 
horizontally, the film moved vertically downward, 
taking some time to come up to the desired speed of 
12.5 in. per sec. Because the film speed at a given 
instant was thus not known accurately, an accurate 
timing record of some sort had to be photographed 
along with the oscilloscope reflection. The chosen 
type consisted of the successive images of a slit which 
were illuminated by the short-duration flashes of a 
strobotron tube driven by an electrically controlled 
fork. 

When operating properly, this system makes a 
faithful record of all intensity changes in the received 
reverberation, since the cathode-ray oscilloscope suf¬ 
fers from no mechanical inertia effects. However, 
since it is impractical to run the camera at speeds 
rapid enough to resolve individual cycles, only the 
time variation of peak reverberation intensity is dis¬ 
cernible on the record. Thus this equipment cannot 
be used to determine the frequency characteristics of 
the reverberation. 

In practice, certain difficulties were experienced 
with this system. For example, when the projector 
and receiver were close to each other, difficulty was 


experienced because of blocking of the receiver ampli¬ 
fier by the received ping, during the period when the 
projector is radiating its sound pulse. If this blocking 
is not eliminated, it leads to what may be described 
as a period of paralysis which lasts for a time after 
the end of the ping. During this period of paralysis 
there is serious distortion of the amplitude of the 
received reverberation. Another problem, also in¬ 
volving blocking effects, was the elimination of 
transients originating during the keying-in of gain 
changes. These transients could not be entirely elimi¬ 
nated in the final versions of this equipment. 

In the derivation of the theoretical formulas for 
reverberation in Chapter 12, it was assumed that the 
projected signal was “square-topped,” or, in other 
words, maintained a constant amplitude for a definite 
interval. No actual ping has this ideal rectangular 
shape, since some time is always required for the ping 
to build up and die away. Figure 2 illustrates the 



Figure 2. Shape of 13 MS ping from QCH-3 pro¬ 
jector. 


shape of a 13-msec ping sent out by a QCH-3 pro¬ 
jector with electronic setup A and recorded by a 
system with flat frequency response. It will be noted 
that a definite time, about 1 or 2 msec, elapses before 
the signal reaches its maximum value. This maximum 
value is the same as the steady state level for a signal 
of indefinite duration, but is not held for long; 7 msec 
from the start of signal emission, the signal level in 
Figure 2 has diminished below its maximum value by 
about 4 db. After 13 msec, the signal dies away; how¬ 
ever, the rate of decay is measurable. 

Other photographs were taken of longer signals, 
more than 100 msec in length. In all these, the signal 
attained its maximum in 1 or 2 msec, fell to 3 or 4 db 
below maximum at 7 msec, and held a fairly steady 
level 3 to 4 db below maximum between 7 and 50 




EQUIPMENT AND FIELD PROCEDURES 


275 



Figure 3. Schematic arrangement of apparatus in system used recently at San Diego (equipment C of text). 


msec. During the interval 50 to 70 msec, the signal 
level gradually rose to its fully steady state value, 
which was maintained for times greater than 70 msec. 

With the QCH-3 equipment A, the smallest re¬ 
cordable reverberation level was usually limited by 
the level of amplifier noise. On occasion, in noisy 
areas, the ambient water-noise level exceeded the 
amplifier-noise level. 

Equipment B 

This equipment was devised for use with the QB 
crystal transducer. Since the QB was used both as a 
projector and as a receiver, the electronic setup had 
to be somewhat different from the equipment de¬ 


scribed under equipment A. A changeover relay had 
to be provided to switch the transducer from the pro¬ 
jector circuit to the receiver circuit. An improved 
power amplifier was built for this system, with the 
result that the projected signal was nearly square- 
topped in form. Since the receiver circuit is not con¬ 
nected while the projector circuit is in operation, no 
blocking during the interval of projection was en¬ 
countered in this system. Even though the receiver 
circuit is not connected during the ping, a record of 
the outgoing ping is obtained on the film which re¬ 
cords the reverberation; this “ping record” is due to 
the electrical cross talk generated in the receiver cir¬ 
cuit by the high voltages in the projector circuit 





























276 


EXPERIMENTAL PROCEDURES 


during the interval of the pulse. Thus, an accurate 
record of the ping length appears on the film. 

With this system, the minimum recordable rever¬ 
beration signal was limited by amplifier noise during 
calm water conditions, and by water noise when the 
sea was choppy. 

Equipment C 

In the early part of 1943, new equipment was put 
into operation by the UCDWR 3 Reverberation 
Group. This equipment can be used with a wide vari¬ 
ety of transducers and was originally provided with 
four distinct frequency channels—10, 20, 40, and 80 
kc. However, any four frequencies between 10 and 80 
kc could be used in the projector circuit by proper ad¬ 
justment of the oscillator resonant circuit. 4 Receiver 
circuit changes to accommodate different frequencies, 
such as provision of properly tuned input trans¬ 
formers and band-pass filters, could also be made 
easily. It appears from later UCDWR memoranda 5 6 
that this system was altered to include a 24-kc chan¬ 
nel and that this channel has actually been used in 
the majority of the reverberation runs made with this 
system. 



10 MS PING IOOMS PING 

Figure 4. Shape of signals sent out by equipment C 

of text. 

The power output with this setup varies with the 
transducer employed. With the JK transducer at 
24 kc, the power output averages about 100 db 
above 1 dyne per sq cm. 6 A block diagram for this 
system, assuming a single transducer unit, is given 
in Figure 3. 

This system differs from systems A and B in a 
number of respects. A major innovation was the use 
of electronic timing circuits to control the ping length 
and keying interval, instead of the complicated me¬ 
chanical motor-driven schemes described previously. 
The changeover relay circuit, used with a single pro¬ 
jector-receiver, is also electronically timed, as are the 
step attenuators which vary the gain in the receiver 


circuit. The pulse projected by this system is practi¬ 
cally square-topped. Figure 4 gives photographs of 
the signal shape for signals of 10 and 100 msec. 

The receiver circuit was specially designed for 
stability in operation. Positions of the gain changes 
were automatically marked on the film by means of a 
flashing lamp. It is clear from Figure 5 that the 
transients during gain changes are not marked 
enough to be troublesome. In this illustration the 
timing trace can be seen at the top of the film; the 
positions of the gain changes are indicated by spots 
on the film below the oscillograph trace, which pre¬ 
cede the actual gain changes by a fixed distance on 
the record. 

Because of its greater convenience, its suitability 
for a large number of transducers, its elimination of 
transients, and the square-topped shape of its emitted 
signal, this system is in many respects a considerable 
improvement over systems .4 and R. 

Equipment I) 

This system is described in references 3 and 7. The 
projector used with this equipment, the EBI-1, 
generated a pressure at 1 yd, on the axis, of 104 db 
above 1 dyne per sq cm. The receiver was a pre¬ 
liminary model, and had a number of faults. 8 Many 
of the basic features of this equipment are similar to 
those in systems A, B, and C. However, the detecting 
mechanism used was not a cathode-ray oscillograph, 
but a Miller galvanometer mounted in a modified 
oscillograph camera. Because the galvanometer could 
not follow 24-kc vibrations, it was operated at 1,000 c 
by heterodyning the received reverberation to this 
frequency. One difficulty with this system is that a 
small percentage change in the i-f oscillator frequency 
(rated at 251 kc) caused a large deviation in the out¬ 
put frequency from 1,000 c, thereby introducing an 
error since the response of the recording galvanometer 
is not wholly independent of frequency. In this sys¬ 
tem, another galvanometer element was used to 
record the current fed to the transducer during each 
ping, while a third galvanometer element was used to 
make the timing marks. 

The Miller galvanometer is naturally resonant at 
2,500 c because of its mechanical inertia, and there¬ 
fore cannot follow the variations in reverberation 
intensity with the detail possible with the cathode- 
ray oscilloscope used in systems .4, B, and C. How¬ 
ever, the Miller galvanometer is convenient to use 
and is certainly capable of following the variations in 
reverberation intensity with sufficient detail for the 




EQUIPMENT AND FIELD PROCEDURES 


277 



I* igure 5. Oscillograph record showing negligible transients produced by receiver amplifier gain changes in equipment C 
(input signal maintained at steady value). 


accurate determination of average reverberation 
levels as a function of time. 

Equipment E 

This equipment used by WHOI is described in 
reference 9. This reference gives only the details of 
the recording system; presumably in most details the 
system does not differ essentially from UCDWR 
systems A to D. One major difference is the introduc¬ 
tion of a logarithmic amplifier which makes it possible 
to record directly in decibels. In the original version 
of this equipment 9 a seismographic galvanometer 
with a linear response up to 70 c was the recording 
device. The deflections of this galvanometer were 
recorded on photosensitive paper. More recently, the 
galvanometer has been replaced by a recording ar¬ 
rangement consisting of a cathode-ray oscillograph 
and a camera. This system has a linear response up 
to about 1,000 c; at higher frequencies the overall 
response, through the logarithmic amplifier, falls off 
rapidly. 

With this WHOI equipment, the averaging pro¬ 
cedure is simplified by superposing on the same film 
the records from a number of successive pulses. When 
the reverberation from a number of pulses is re¬ 
corded on one film in this manner, the average rever¬ 
beration curve can be drawn by eye through the 
densest portions of the trace. Since the film records 
the reverberation in decibels, the resulting plot is the 
desired curve of average reverberation level versus 
time. A sample record, showing the superposition of 
reverberation from 12 successive pulses recorded with 
the seismographic galvanometer, is shown in Figure 
6. In this illustration, time increases from right to 
left; the particular features of the galvanometer used 
by WHOI made it more convenient to present the 
data in this way. 

Table 1 summarizes some of the important inior- 
mation concerning the equipment used in various 
reverberation studies at UCDWR. Most of the items 
in the table are self-explanatory. The letters A to D 


refer to the electronic setups described in equipment 
designations; and the figures in parentheses next to 
the transducer designations in the column labeled 
“Transducer references” tell where detailed descrip¬ 
tions of these transducers can be found in the 
bibliography. The column labeled “Reference” tells 
where the results obtained with the indicated equip¬ 
ment are discussed. 


Table 1. Equipment used in reverberation measure¬ 
ments. 


Reference 

Transducer 

references 

Frequency 
in kc 

Electronic 

setup 

1 

QCH-3 (O 

24 

A 

1 

QB «> 

24 

B 

4, 10 

GB 

10 

C 

4, 10 

GA.GB 

20 

C 

4, 10 

GA (10 - ll) 

40, 80 

c 

7 

EBI-1 <“> 

24 

D 

5 

JIv < 12 > 

24 

C 


13.1.2 Calibration of Projector 
and Receiver 

In order to properly interpret the recorded values 
of reverberation, and to convert these recorded ampli¬ 
tudes to reverberation levels, it is necessary to know 
the values of the projector output F and the receiver 
sensitivity F'. These quantities, which occur in the 
theoretical formulas of Chapter 2, depend not only 
on the type of transducer, but also on the electronic 
equipment used. The procedures used in the determi¬ 
nation of F and F' in the field are called “calibra¬ 
tion procedures.” 

The projector is calibrated by measuring the pro¬ 
jector output with a standard hydrophone whose 
response is stable. If an auxiliary projector with 
stable power output is available, F' can be deter¬ 
mined by measuring the output of the receiver when 
exposed to a pulse from the standard projector. If the 
output of the auxiliary projector is not accurately 










278 


EXPERIMENTAL PROCEDURES 



Figure 6. Sample record from Woods Hole reverberation camera (reverberation from 12 successive pulses superposed). 


known, the auxiliary projector itself may be cali¬ 
brated with the aid of the standard receiver, and then 
used to calibrate the receiver. The echo received from 
a sphere of known target strength at a known dis¬ 
tance has been used by WHOI to determine the 
product FF'. Present practice at both laboratories is 
to make a calibration at least once every working 
day, whenever possible. 6 

Although calibration is simple in principle, experi¬ 
ence has shown that there is likely to be considerable 
inaccuracy in all projector and receiver calibrations. 
At UCDWR, it was found that the values of F and F' 
determined by calibration procedures may change 
unaccountably with time, sometimes changing by 
nearly 10 db from day to day, and by somewhat 
lesser amounts during a single day. 6 Some method 
for detecting calibration errors in the field is desirable 
since these errors are reflected as errors in the rever¬ 
beration levels inferred from the measured intensities. 

13.1.3 Typical Reverberation Records 

Most of the reverberation data obtained by 
UCDWR are in the form of oscillograms on 35-mm 
motion picture film. Figure 7 shows a sample record 
of the reverberation from three successive pings sent 
out at 8-sec intervals. For convenience in display, 
each reverberation record was cut into three sec¬ 
tions, as shown in the illustration; the three A ’s make 
up the first record; the three B 's the second; etc. The 
marks on the upper edge of each record give the time 
scale; these marks are 2.5 msec apart. The point a 
represents the emission of the signal; b the onset of 
reverberation with transients caused by the opera¬ 
tion of the changeover relay; and c, d, e, places where 
the gain was automatically increased by the atten¬ 
uators. At /, the reverberation has decreased below 


background noise. The film speed (12.5 in. per sec) is 
high enough to show considerable fine structure in 
the reverberation. However, it is not high enough to 
resolve individual cycles; thus the trace shown in 
Figure 7 represents the envelope of the received 
reverberation. 

The records shown in Figure 7 are quite typical and 
illustrate some of the statements which have been 
made in this volume about the behavior of reverbera¬ 
tion. The recorded amplitude at a given time past 
midsignal is not constant from record to record, even 
though these pings were sent out and the reverbera¬ 
tion was recorded under the same adjustments of the 
experimental apparatus. In general, however, when¬ 
ever reverberation measurements are made, there 
are major features which persist from record to rec¬ 
ord. One such characteristic is the point of onset of 
bottom or surface reverberation. Another is the in¬ 
variable tendency of reverberation of a given sort 
(volume, surface, or bottom) to decrease with in¬ 
creasing time, as is predicted by the theoretical 
formulas of Chapter 12. This decrease makes neces¬ 
sary the provision in the system of gain changes at 
points such as c, d, and e; without these gain changes 
it would be impossible to record all the reverberation 
at measurable amplitudes. Occasionally, successive 
reverberation records show a systematic increase at 
certain points. These increases can usually be cor¬ 
related with the calculated increase due to the onset 
of surface or bottom reverberation; sometimes they 
are ascribed to the existence of local regions of high 
scattering strength. 

13.2 ANALYTICAL PROCEDURES 

After the field work is done, the films containing 
the received reverberation records are taken to the 




ANALYTICAL PROCEDURES 


279 



Figure 7. Oscillograph records of reverberation from three successive pings. 


laboratory to be analyzed and averaged. These rec¬ 
ords are divided into sets, each consisting of records 
taken within a short space of time under similar 
conditions. The reverberation measurements making 
up a set are then averaged, and the resultant averages 
are supposed to represent the expected reverberation 
under the known external conditions for the set. 
Obviously the averages cannot be computed for every 
time instant after midsignal. Times are chosen which 
are spaced closely enough so that the major system¬ 
atic changes in reverberation level will be evident.® 
At UCDWR, two methods of averaging have been 
used: “point method” and “band method.” 

The point method of averaging is to select a set of 
points, such as 1, 2, and 3 in Figure 7; measure the 
amplitude at these points; repeat for all records in the 
set (usually from 5 to 10); make proper allowance for 
gain changes and projector-receiver calibration; con- 


a In selecting and manipulating the data, places on the 
records where obviously extraneous noise showed up have 
customarily been rejected. Examples of extraneous noises are 
pings from destroyers, echoes from porpoises, and bursts of 
ship noises. 


vert the amplitudes to decibels above the chosen ref¬ 
erence level; and finally, plot the resulting average 
reverberation levels as a function of time or range. 
Outstanding features, such as reverberation from the 
bottom or from a suspected deep scattering layer, can 
be emphasized by choosing many points in their 
vicinity on the records; and uneventful portions of 
the record can be passed over with but one or two 
points to set the general level. Usually the points 
chosen were spaced so as to give equal intervals on 
logarithmic coordinate paper. 

The alternative method, the band method, was 
introduced because of the considerable difficulty in¬ 
volved in computing an accurate average with the 
point method. On some records, the amplitudes are 
changing very rapidly close to the predetermined 
point where the amplitude is to be read; and to 
measure these amplitudes accurately it is necessary 
to look at the records very closely with appropriate 
viewing devices. This procedure is both time-con¬ 
suming and hard on the eyes, especially when the 
amplitude at the predetermined time is small. 

In the band method, a set of points is chosen as in 





280 


EXPERIMENTAL PROCEDURES 


the point method. But the amplitude measured is 
not the amplitude at that point, but the greatest 
amplitude in a band three ping lengths long and 
centered at the designated point. Corresponding 
amplitudes are measured for all similar records; al¬ 
lowance is made for gain changes and projector-re¬ 
ceiver calibration; finally, after converting to deci¬ 
bels, the average reverberation levels are plotted as a 
function of time. As a procedure for plotting rever¬ 
beration data, the band method seems definitely 
superior to the point method. The amplitude cox- 
responding to a particular point is much easier to 
obtain with the band method, since it is simpler to 
pick out the maximum amplitude in an interval than 
to measure the amplitude at a predetermined point. 
Also, amplitudes obtained with the band method 
show much less fluctuation than amplitudes obtained 
with the point method; an analysis of reverberation 
i - ecoi - ds consistently showed a standard deviation of 
amplitude for the band method of less than 50 per 
cent of the standard deviation for the point method. 1 

It is difficult to see the exact significance of the 
averages obtained with the band method. Certainly 
the band method does not closely resemble the avei- 
aging method which was the basis for the theoretical 
formulas of Chapter 12; the point method, on the 
other hand, does resemble it. Thus, in order to com¬ 
pare the observational resixlts obtained with the 
band method with theoretical expectations, the sim¬ 
plest procedure is to coi’rect the band method results 
to what would have been obtained had the point 
method been used. The amount of this correction was 
determined expei’imentally by comparing the results 
for many records pi'ocessed by both the point and 
band methods. Except at very short ranges, it was 
found that the band method gives results which 
average quite consistently 7 db greater than results 
obtained with the point method. Subtraction of 7 db 
from the band method results thus gives average 
reverberation levels which are comparable with the 
theoretical expectations of Chapter 12. At very short 
ranges, on the other hand, the reverberation is chang¬ 


ing so l'apidly that the band method does not give 
sufficient detail and does not show any consistent 
relationship to the point method. 

Some more details of the piesent UCDWR analyt¬ 
ical pi-ocedure may be of interest to the reader. The 
individual records are analyzed by placing the films 
in a viewer against a graph paper background. Verti¬ 
cal and hoi-izontal distances on the film can be meas¬ 
ured by counting squares on the graph paper, which 
is usually ruled in millimeters. In analyzing a recoi-d, 
the analyst first measures the ping length in terms of 
squares on the graph paper and converts this to 
milliseconds by comparing millimeters and the dis¬ 
tance between points on the timing trace. The num¬ 
ber of timing marks in a fixed film length gives the 
film speed from which a scale of l’ange from midsignal 
may be constructed. This range scale is set up next to 
the film in the viewer. At ranges greater than 250 yd, 
the band method is used to detemxine the amplitude 
at the designated range. At ranges of 100 yd and 
250 yd, however, the point method is used because 
at these short ranges, as explained previously, the 
l-everberation is changing too l-apidly for the band 
method to give accurate results. 

Despite the simplifications introduced by the band 
method of averaging, the analysis of a set of UCDWR 
reverberation l-ecords is an arduous and time-con¬ 
suming pi-ocess. In the WHOI system E, the final 
plot of avei'age reverberation level against range can 
be obtained immediately from the photographic 
paper, by drawing a curve through the densest area 
on the superposed reverberation traces. This system 
is highly convenient for recording and plotting aver¬ 
age reverberation levels; but it does not permit any 
detailed measurement of x-everberation fluctuation. 
Probably the best system for recording reverberation 
would combine the advantages of both the UCDWR 
and the WHOI types. This equipment would make a 
permanent record of all fluctuation on one l'ecoi'ding 
element, while on the other recoi’ding element a 
smoothed trace would be made from which the final 
reverberation levels could be readily obtained. 



Chapter 14 


DEEP-WATER REVERBERATION 


I t is convenient to begin the study of observed 
reverberation levels by describing the experi¬ 
mental observations in deep water. In deep water it 
is usually possible to ignore bottom reflections, 
thereby facilitating comparison of the experimental 
results with the theoretical formulas of Chapter 12. 
Also, in deep water, it is frequently possible to elimi¬ 
nate surface scattering and reflections, by directing the 
beam downward at some angle. When reverberation 
from the surface and bottom is effectively eliminated, 
the received reverberation can assuredly be called 
volume reverberation. The first section of this chap¬ 
ter describes the experimental facts about volume 
reverberation, as determined by such unambiguous 
experiments. 

In ordinary echo ranging, with the main trans¬ 
ducer beam horizontal, part of the received reverbera¬ 
tion is surface reverberation and part volume rever¬ 
beration. It is not easy to make a clear distinction be¬ 
tween these two components on observed reverbera¬ 
tion records. The distinction between the two is 
usually made by comparing the measured levels ob¬ 
tained with horizontal beams with the levels observed 
in unambiguous volume reverberation experiments, 
and also by observing the dependence of the rever¬ 
beration levels on sea state. The observed levels with 
horizontal transducer beams are described in the 
second section of this chapter. 

14.1 TRANSDUCER DIRECTED 

DOWNWARD 

The experimental method for eliminating surface 
reverberation in deep water has usually been to point 
a highly directional transducer downward, away from 
the surface. In this way the main transducer beam 
does not strike the surface, and the observed rever¬ 
beration levels are then assumed to be due to volume 
reverberation. Of course this assumption requires 
verification, since in the absence of any information 
about the relative values of the surface and volume 


backward-scattering coefficients, it is not possible to 
know in advance how much directivity is necessary 
to definitely eliminate surface reverberation. How¬ 
ever, it may be accepted as a working hypothesis that 
pointing the main transducer beam down 30 de¬ 
grees, or more, does eliminate surface reverberation, 
for standard 24-kc echo-ranging gear. It will be seen 
later that surface reverberation levels are not usually 
high enough to contribute to the received reverbera¬ 
tion under these circumstances. The following sub¬ 
sections describe the various experimental studies of 
volume reverberation which have been carried out 
in this manner. 



Figure 1 . Volume reverberation levels showing in¬ 
verse square range dependence. 


14.1.1 Dependence on Range 

According to Chapter 12, equation (22), if the 
volume scatterers are uniformly distributed, and if 
the transmission anomaly terms —2 A 4- A\ in equa¬ 
tion (22) can be neglected, then the volume rever¬ 
beration intensity should be inversely proportional 
to the square of the range, or in other words, the 
reverberation level should decrease 20 db with a ten¬ 
fold increase in time. As an example of this depend¬ 
ence, we may refer to Figure 1, which is a plot of data 
obtained on June 3, 1942. 1 The QCH-3 transducers, 
projector and hydrophone, were lowered to a depth 
of 60 ft and tilted downward 60 degrees. The water 
depth was 600 fathoms and the surface was moder- 


281 












































282 


DEEP-WATER REVERBERATION 



0.01 0.05 0.1 0.5 1 5 10 


TIME IN SECONDS 


Figure 2. Volume reverberation 

ately calm with the wind velocity averaging 10 mph, 
and long low ground swells but no whitecaps. A signal 
length of 10 msec was used. Twenty records were 
filmed, measured, and averaged, to give the points 
shown in Figure 1. It is seen that the experimental 
data fit fairly well the straight-line dependence of R' 
on log r which is predicted, if all quantities except R’ 
and r are constant, by equation (24) of Chapter 12. 

In practice, volume reverberation runs usually 
show even worse agreement with this simple linear 
range dependence than do the points shown in 
Figure 1. In the first place, it is known from trans¬ 
mission measurements that the transmission anomaly 
terms can rarely be neglected at ranges greater than 
1.000 yd (see Chapter 5). Thus the inverse square 
dependence can be expected only at relatively short 
ranges. In addition, there is no real reason to expect 
the volume scatterers to be uniformly distributed in 
the ocean. However, the fact that an approximately 
inverse square dependence has been observed in at 
least a few cases is evidence that our fundamental 
assumptions about volume reverberation are not 
altogether wrong. In general, volume reverberation 
tends to decrease rapidly with increasing range, in at 
least qualitative agreement with equation (24) of 


OCEAN SURFACE 



C hapter 12. However, the detailed dependence on 
range is frequently observed to be very different from 
the simple form of that equation; often the depend¬ 
ence of R’ on log r is not linear, and when it is linear, a 
slope of exactly —20 is quite unusual. Such observa¬ 
tions are described in the following subsection. 

14.1.2 Dependence on Depth 

Measurements off San Diego with the transducer 
pointed downward have frequently shown sudden 
increases in reverberation level which seemingly 
could only be explained by assuming that in certain 
deep layers of the ocean the backward volume-scat¬ 
tering coefficient was much larger than at other 
depths. Figure 2 was drawn from data obtained on 
July 28, 1942. The QB transducer was pointed down¬ 
ward at an angle of 49 degrees relative to the hori¬ 
zontal, in 660 fathoms of water. Ten records were 
averaged to give the points in this figure. At the re¬ 
verberation range indicated by .4 in the illustration 
there is a sharp rise of more than 10 db in reverbera¬ 
tion level. A comparison with equation (24) of Chap¬ 
ter 12 makes it seem necessary to ascribe this rise to 
an increase in the backward scattering coefficient m\ 























































TRANSDUCER DIRECTED DOWNWARD 


283 


certainly it does not seem possible that any change in 
transmission anomaly could be sufficiently sudden to 
account for the rise. The geometry of the experiment 
is shown in the small box of Figure 2, on the assump¬ 
tion that the ray paths are approximately straight 
lines. The peak at *4 occurs at a time 0.5 sec after the 
ping. This corresponds to a reverberation range of 
400 yd; thus, the layer of high scattering power must 
have been centered at a depth of 400 yd X cos 41°, 
or about 900 ft. The thickness of the layer, as esti¬ 
mated from the thickness of the bulge at .4 in Figure 
2, was not less than 500 ft. The large increase in re¬ 
verberation level at B corresponds to the point at 
which the beam strikes the bottom. The rise at C 
in Figure 2 could have resulted from scattering of 
bottom-reflected sound by the deep scattering layer. 
It could also have resulted from sound which was 
scattered from the bottom toward the surface, re¬ 
flected from the surface back to the bottom, then 
scattered from the bottom back to the transducer. 
These various possible paths are shown in the small 
box in Figure 2. 

Another record of the many which show the pres¬ 
ence of a deep scattering layer is one made August 5, 

RANGE IN YARDS 



Figure 3. Volume reverberation levels with deep 
scattering lawyer. 


1942. The data, plotted in Figure 3, were obtained 
with the QB transducer pointed vertically downward 
in 650 fathoms of water. Figure 3 is an average of 10 
pings each 12 msec long. This experimental curve has 
several important features. The first portion of the 
curve decreases as 20 log t, indicating uniform distri¬ 
bution of scatterers to a depth of about 500 ft. A 
deep layer of high scattering power is evident in the 
vicinity of A in Figure 3; this layer appears to have 
a mean depth of 1,000 ft and a thickness of about 
750 ft. At the position of highest scattering power 
within the layer, the volume-scattering coefficient is 
very much greater than its value in the body of the 


ocean above the layer. If we use equation (24) of 
Chapter 12 to estimate 10 log m, assuming that the 
transmission anomaly terms are small, then 10 log m 
at A is 16 db greater than 10 log m at points on the 
line denoting inverse square decay; in other words, 
in at A is 40 times as great as m at points in the first 
500 ft of the ocean. Once the beam is out of the layer, 
the reverberation level falls off abruptly. At a depth 
of 2,250 ft, the calculated value of 10 log m, neglect¬ 
ing the transmission anomaly terms in equation (24) 
of Chapter 12, is 20 db down from the value of 10 log 
m in the first 500 ft. This difference could not be ac¬ 
counted for by ordinary values of the transmission 
anomaly terms — 2A + A x . It is possible (though not 
likely) that the sound suffers an abnormally high 
transmission loss in its two-way passage through the 
high scattering layer, or there may actually be a 
layer of low scattering power at the 2,250-ft depth. 
Echoes from the bottom are noted at B, D, and E in 
Figure 3. The distance the sound which produces a 
reverberation peak has traveled can be estimated by 
noting the time at which the peak appears; it is easily 
seen that the sound producing the peak at D has 
gone from the transducer to the bottom, back to the 
surface, then to the bottom again, and finally back 
to the transducer. By a similar computation the peak 
at C is seen to be sound which traversed one of the 
following two paths: (1) scattered by the layer up 
to the surface, reflected from the surface to the bot¬ 
tom, and then returned to the transducer, (2) re¬ 
flected from the bottom up to the surface, reflected 
back toward the bottom, and then scattered back 
to the transducer from the deep layer. 

Not all scattering layers are at great depths. For 
example, a scattering layer at a depth of about 200 ft 
is evident in the reverberation curve of Figure 4. 
These records were taken with the sound beam di¬ 
rected vertically downward, and with a ping length 
of 10 msec. Occasionally both shallow and deep scat¬ 
tering layers are present in the ocean simultaneously. 
An example is given in Figure 5, which is made up of 
reverberation from 8-msec pings projected at a de¬ 
pression of 60 degrees below the horizontal in 620- 
fathom water. In that figure, three scattering layers 
are noted, at A, B, and C. The layer A is at a depth 
of about 100 ft, B at about 600 ft, and C at about 
1,000 ft. 

Some of these deep scattering layers appeared to 
persist for relatively long periods of time. In the same 
area of operation as that for Figure 2, deep scattering 
layers were observed at about the same depth over a 














































284 


DEEP-WATER REVERBERATION 


< 

q: 

uj 

00 


-80 


-100 


-120 


40 


80 


RANGE IN YARDS 
400 


800 


4000 


8000 


-140 










































- 

• 

•- 

A\ 

E 

/E 

VIF 

R 

IF 

1GE OF 10 R 

ICAL CURVE 

EC0RDS 
































✓ 

*- 


• 




























V 





























--• 


-• 

- 

-• 

■l 




N0IS 

E L 

EVE 

L 





0.01 


0.05 


0.1 


0.5 

TIME IN SECONDS 


10 


TEMPERATURE IN DEGREES F 



Figure 4. Volume reverberation levels with shallow scattering layer. 


period from July 9 to August 5, 1942. On the other 
hand, on June 16 and 17, a layer was observed at 
1,200 ft; but a week later no such layer was detected. 
Thus the observations indicate that deep scattering 
layers, in a given area, may sometimes appear and 
disappear, and at other times persist for periods as 
long as a month or even longer. Just what these deep 
scattering layers consist of is not known; they may, 
for example, be concentrations of fish, bubbles or 
plankton. 8 The layer of Figure 4 occurs at the same 
depth as does a temperature inversion on the bathy¬ 
thermograph trace shown in the insert of Figure 4. 
On the other hand, no inversion is noticed at the 
depth corresponding to .4 in Figure 5. 

14.1.3 Dependence on Frequency 

An extensive series of measurements of volume 
reverberation in deep water, 2 at frequencies of 10, 20, 

“ The observations reported in this chapter were made dur¬ 
ing daylight hours. More recent studies show evidence of 
diurnal migration of the deep scatterers and lend support to 
the theory of biological origin. 


40, and 80 kc have been made by UCDWR. These 
measurements, described later, were made in water 
depths ranging from 660 to 1,950 fathoms, in the 
months of January and February 1943. The area of 
observations extended southwest of San Diego to 
Guadalupe Island, which is about 250 miles from 
San Diego and 200 miles off-shore. The various posi¬ 
tions at which observations were taken are marked 
by roman numerals in Figure 6. 

Deep scattering layers of the type discussed previ¬ 
ously were observed on this cruise. Figures 7 and 8 are 
plots of typical reverberation records obtained at 
three positions shown in Figure 6. These data were 
obtained with the transducers directed vertically 
downward, sending out 10-msec pings at the four 
frequencies 10, 20, 40, and 80 kc. Each point on the 
curves for positions III and \ III is an average of 5 
pings, while points on the curves for position IX are 
an average of 25 pings. It is evident from Figures 7 
and 8 that the effective depth of the deep scattering 
layer does not seem to depend on frequency. This 
fact is shown somewhat better in Figure 9, which is a 





















































TRANSDUCER DIRECTED DOWNWARD 


285 


RANGE IN YARDS 

4 0 80 400 600 4000 5000 
































A 











• 

• 


A 

E 

VE 

M 

RAGE OF 

PIRICAL Cl 

10 REC 

JRVE 

;ord 








•- 


-• 










s ■ 












\ 

\ 

k fl 


B 

• 

























i 

/ 

/ 

/• 

/ 

J * 

/ 

/ 

\ 

\ 

• 

f 

h« 





























-< 

>- 

« 


N 

_ 

DISE 

LE 

VE 

L 

_ 





0.01 0.03 0.1 0.5 1 5 10 

TIME IN SECONDS 


Figure 5. Volume reverberation levels 

plot of the estimated depth of the deep layer ob¬ 
served for each position and frequency along the line 
connecting positions III and VIII. Figure 9 illustrates 
the persistence of the layer throughout the area of 
observations. 

According to equation (24) of Chapter 12, it should 
be possible to determine log m from the experimen¬ 
tally observed reverberation levels, provided the 
values of the transmission anomaly term —2.4 + Ai 
are known. Since horizontal velocity gradients in the 
ocean are usually negligible, refraction can be neg¬ 
lected in measurements with a directional transducer 
pointed vertically downward, and A\ can thus be set 
equal to zero. Furthermore, if the acoustic properties 
of the ocean do not change much with increasing 
depth, the transmission anomaly resulting from ab¬ 
sorption and scattering should be a linear function of 
range (see Section 5.2.2 of Part I). In other words, if 


TEMPERATURE IN DEGREES F 



with scattering layer at several depths. 

the ocean is approximately homogeneous, the term 
— 2 A + A i in equation (24) of Chapter 12 should 
equal — 2or/l,000 where r, the range of the rever¬ 
beration in yards, is equal to the depth of the scat¬ 
tered giving rise to the reverberation. It follows 
that if the “uncorrected” scattering coefficient M is 
determined from the equation 

R'(t ) = 10 log y + 10 log M - 20 log r + ./„, 

then 

2ar 

10 log M = 10 log tn — --•• (1) 

& - 1,000 

In equation (1) m, the true value of the scattering 
coefficient, is constant if the properties of the ocean 
do not change with depth. Thus, for a homogeneous 
ocean, with m and a constant with depth, a plot of 
10 log M against depth on a linear scale should be a 



























































286 


DEEP-WATER REVERBERATION 



119° 118° 117° 116° 115° 


LONGITUDE 

Figure 6. Locations where reverberation was meas¬ 
ured in 1943 cruise. 

straight line. The slope of this line will determine a, 
the attenuation coefficient in decibels per kiloyarcl; 
and the intercept of the line at zero range will deter¬ 
mine the value of the true scattering coefficient m. 

However, the very existence of the systematic in¬ 
crease in reverberation levels at about 1,000 ft ob¬ 
served in Figures 7 and 8 means that the ocean is 
probably not homogeneous with depth; thus a 
straight-line dependence of 10 log M on depth could 
hardly be expected in this experiment. Figure 10 is a 
plot of the mean values of 10 log M for the nine sets 
of records observed in the period January 17 to 20, 
1943, at the positions shown in Figure 6. It is obvious 
from Figure 10 that even if the points in the deep 
layer between 1,000 and 1,500 yd are ignored, no 
good fit to the data could be obtained with a straight 
line. 

The failure to obtain a straight-line dependence in 


Figure 10 means that either m or a, or both, change 
with depth. It is possible to obtain further informa¬ 
tion from Figure 10 by comparing the dependence on 
depth at different frequencies. From equation (1), for 
any two frequencies/i and/ 2 , 


10 log 


M(h) 

M(h) 


= 10 log 


m(fi) 

m(J 2 ) 


~ 2[«(/i) 


a (/a)] 


1,000 ’ 


( 2 ) 


If the variations in m are caused only by changes in 
the number of scatterers per unit volume, then 
m(fi)/m(f 2 ) should be independent of depth. Thus, 
if the attenuation is independent of depth, 10 log 
M(Ji)/M{f 2 ) in equation (2) should be a linear func¬ 
tion of depth in these runs with the transducer di¬ 
rected vertically downward. Figure 11 is a plot of 
this ratio against depth, from the data of Figure 10, 
for the six pairs of frequencies involved. Only three 
of the ratios are independent; the other three can be 
calculated from the first three. All the ratios are 
shown in Figure 11 for comparison. Although most 
of the graphs show general tendencies to slope in the 
direction of increasing attenuation at higher fre¬ 
quencies, systematic deviations from the straight 
line predicted by equation (2) are noted. It appears 
then that either the kind of scatterer changes with 
depth or the attenuation coefficient varies with 
depth. 

At distances less than 250 ft, attenuation is small, 
even at 80 kc. Thus the scattering coefficients in the 
upper 250 ft of the ocean can be computed from 
vertical reverberation runs without knowledge of the 
attenuation coefficient. Mean values of 10 log M ~ 
10 log m, averaged over seven depths between the 
surface and 250 ft, are plotted in Figure 12 as a func¬ 
tion of frequency, for each of the nine positions of the 
sending-receiving ship during January 1943 (Figure 
6). The solid lines are empirical curves and the dashed 
lines represent a theoretical relationship discussed 
later. The shapes of the empirical graphs for the 
different positions bear little resemblance to each 
other. However, the two curves for position III, 
which represent data taken 20 hours apart, reproduce 
each other almost to within sampling error. The 
curves for positions I and VIII, which were close to¬ 
gether in space but separated by 72 hours in time, 
are also nearly identical. If these resemblances are 
not accidental, they suggest that position is a more 
important factor than time in determining the value 
















TRANSDUCER DIRECTED DOWNWARD 


287 



Figure 7. Observed volume reverberation levels versus scattering depth; GB units; sound beam vertical. 


of the volume-scattering coefficient. It also appears 
from Figure 12 that the scattering coefficient is not 
affected in the same way at all frequencies by changes 
in position. These results, if verifiable, also substanti¬ 
ate the hypothesis that, volume reverberation is not 
an intrinsic property of water as such, but results 
from scatterers in the ocean whose number and type 
are affected by oceanographic and climatic condi¬ 
tions. Certainly, if reverberation were a property of 


water as such, it is difficult to see how small changes 
in position could result in the different shapes ob- 
servable.in the curves of Figure 12. 

Figure 13 shows the mean values of 10 log M 
averaged at each frequency over all the positions of 
Figure 12 and plotted as a function of frequency. 
The vertical lines in Figure 13 represent mean devia¬ 
tions from this average of the values plotted in 
Figure 12. Figure 13 shows that, on the average, there 

































































































































































































288 


DEEP-WATER REVERBERATION 






REVERBERATION LEVEL R', IN OB 

Figure 8. Observed volume reverberation levels versus scattering depth; GB units; sound beam vertical. 


is a slight increase in 10 log M with frequency. This 
systematic increase is small compared to the irregular 
variation from position to position, but according to 
reference 2, the observed trend is considerably larger 
than the sampling error of the measurements, and 
also somewhat larger than the errors which could be 
introduced by calibration. The straight line shown in 
Figure 13 was fitted by least squares; its slope indi¬ 
cates that M ~ tn increases as the 0.9 power of the 
frequency. It seems safe to say that the results of 
reference 2 do not exclude the possibility that on the 
average the scattering coefficient is independent of 
frequency. They admit the possibility also that m 
may vary as the second power of the frequency but 
not that it varies as the fourth power of the frequency. 
The lines 

m = kf 4 (3) 

are drawn in Figures 12 and 13 for comparison. This 
fourth-power dependence of the scattering coefficient 


on frequency is known as Rayleigh’s scattering law 
and is true for scattering from particles whose di¬ 
mensions are small compared to the wavelength of 
the scattered sound. 3 

14.2 TRANSDUCER HORIZONTAL 

That short-range reverberation with horizontal 
pings is often due primarily to scattering from the 
surface of the sea has been amply demonstrated by 
experiment. Reverberation intensity has been meas¬ 
ured first with the sound beam directed horizontally, 
and then with it directed vertically downward. In the 
first transducer position, surface scatterers are ir¬ 
radiated by much of the central portion of the beam; 
in the second position, they are strongly discrimi¬ 
nated against by the directivity of the transducer. 
When the experiment is performed in a choppy sea 
with whitecaps, the horizontal reverberation is many 
decibels higher than the vertical reverberation at 




























































































































































































TRANSDUCER HORIZONTAL 


289 


POS POS POS POS POS POS 



Figure 9. Depth of deep scattering layer at various positions. 


short ranges. Furthermore, the difference is usually 
much greater in rough seas than in calm seas. Of 
course, it is possible that the volume reverberation 
obtained with the beam directed downward is less 
than the volume reverberation with horizontal beams. 
However, it is pointed out below that observed 
values of 10 log m with horizontal beams are at most 
only about 6 db greater than values of 10 log m meas¬ 
ured with vertical beams. Thus, the conclusion is in¬ 
escapable that in rough seas the short-range rever¬ 
beration from horizontal pings is surface reverbera¬ 
tion, that is, reverberation caused by scatterers near 
the surface of the sea whose number and strength 
are a function of sea state. 


14.2.1 Dependence on Range and 
Oceanographic Conditions 

Analysis of reverberation records clearly shows 
that the range dependence of surface reverberation is 
itself a marked function of such oceanographic 
parameters as sea state and temperature gradients. 
For this reason, it is convenient to treat the effects of 
all these variables together. The following section 
summarizes the observational results of studies of 
surface reverberation, which indicate that surface 
reverberation tends to fall off with increasing range 
much faster than predicted by the simple theory of 
Chapter 12, and that the rate of decay increases 

































































SCATTERING DEPTH IN FEET SCATTERING DEPTH IN FEET 


290 


DEEP-WATER REVERBERATION 



Figure 10. Mean value of 10 log M for all positions 
in 1943 cruise. 


rapidly with increasing sea state. A later section en¬ 
titled “Possible Explanations” advances some fairly 
plausible qualifications of the simple theory which 
may explain away, in part, the seeming discrepancies 
between theory and observation. 

Experimental Results 

Figure 14 illustrates the results of one experiment 
comparing the reverberation with horizontally and 
vertically directed beams. At the time of the experi¬ 
ment, the sea surface was confused, with whitecaps 
present and with a wind velocity of 17 mph. A com¬ 
parison of the two reverberation level curves shows 
that for times up to 0.1 sec the horizontal reverbera¬ 
tion is more than 20 db above the vertical reverbera¬ 
tion; for times between 0.1 and about 0.4 sec, it is 
more than 10 db above; and for times greater than 
0.4 sec it is less than 4 db above. Two conclusions are 
obvious from the figure. One is that on the day of the 
experiment there was a surface layer of scatterers 
which was very different in nature from the scatterers 
in the ocean body. The other is that the reverberation 
due to surface scatterers decays more rapidly than 
the reverberation due to volume scatterers. The pres¬ 
ence of a deep scattering layer can also be noted at B 
in Figure 14, and also a shallower scattering layer 
at A. 

According to equation (39) of Chapter 12, the sur¬ 
face-reverberation intensity at short range, where 2 A 
in equation (39) can be neglected, should be propor¬ 
tional to the inverse cube of the range, provided 



Figure 11. Variation with depth of ratio of scattering coefficients. 

















































































TRANSDUCER HORIZONTAL 


291 



o 

o 









—i 







s' 

S n = 4 







> 


7 













P< 

DS 

V 

1 


10 20 40 80 

FREQUENCY IN KC 













s 

S 

y n = 4 








^ - 

> 












PC 

)S 

III 





/ 

^- 








-- > 

s 

S n = 4 






















PC 

LJ 

)S 

V 




10 20 40 80 


Figure 12. Variation of 10 log M with frequency (mean values of 10 log M for depths less than 250 feet). Positions III 
and IIIo refer to measurements made at Position III on two separate days. 


s 

O 

o 

-I 

o 



FREQUENCY IN KC 


Figure 13. Variation of scattering coefficient with 
frequency (mean values of 10 log M at all nine positions 
for depths less than 250 feet). 


RANGE IN YARDS 
40 80 400 800 



10 log m '/2 and J s {6) in equation (39) are also inde¬ 
pendent of range. This simple inverse cube depend¬ 
ence is observed only rarely. Figure 15 shows the 
reverberation intensities observed on May 8, 1942, 
with the QCH-3 transducers. On this date ground 
swells were long and low, and a few whitecaps were 
forming. The QCH-3 transducers were at a depth 
of 20 ft with their long dimensions horizontal and 
with the transducer axes parallel to the sea surface. 


Figure 14. Comparison of reverberation from hori¬ 
zontally and vertically directed beams. 

In this position the QCH-3 transducers are prac¬ 
tically nondirectional in the vertical plane, so that 
in equation (41), Chapter 12, the correction factor 
bid — £,0 )b'(d — £,O)/cos0 is very nearly unity at 
all angles of importance. Since this correction factor 
is the only part of J s id) which can depend on range, 























































































































































































































292 


DEEP-WATER REVERBERATION 


it is clear that in these experiments J,(0) was inde¬ 
pendent of range. With negligible *1 and constant m' 
and J s (0), the theoretical equation for surface rever¬ 
beration, equation (43), Chapter 12, leads to a 
straight line with a slope corresponding to inverse 
third-power decay. This simple reverberation decay 
is indicated by the solid line in Figure 15. The points 
in Figure 15 are the averages of 36 pings each 8 msec 
long and agree fairly well with the theoretical 
straight line. 


RANGE IN YARDS 



£ 0.01 0.05 0.1 0.5 1 


TIME IN SECONDS 

Figure 15. Surface reverberation levels showing sim¬ 
ple inverse cube dependence. Projector and receiver 
effectively nondirectional in the vertical plane. 


800 yd the observed points in Figures 15 and 16 are 
in close agreement. This agreement was not to be 
expected if the scattering coefficient of the surface 
did not change with time; it is easily verified that 
the values of ./«(£?) are quite different for the hori¬ 
zontal and vertical orientations of the QCH-3. b Thus 
the agreement at ranges greater than 80 yd between 
the observed points in Figures 15 and 16 means that 
the value of the surface scattering coefficient must 
have changed during the interval between the two 
experiments. The value of 10 log m '/2 estimated 
from Figure 15 is about 5 db greater than the value 
estimated from Figure 16. 


CD 

O 

z 

~a. 


UJ 

> 

UJ 


< 

<r 

UJ 

CD 

cr 

UJ 

> 

UJ 

<r 


RANGE IN YARDS 
40 80 400 800 


- 

















X 




r 

« 









































* ^ 






• A V 
-TH 

CU 

ERA 

E0R 

RVE 

GE 

ETI 

FO 

0 

CA 

R 

iL 

fH 

3( 

e 

E 

u 

SE 

R 

R 

ECORDS 
FACE RE 
5IRECTIC 

IVEF 

)NAL 

IBE 

lit 

Rf 

JIT 

kT 

S 

« 

0 

• 

N 

L. 


0.01 0.05 0.1 0.5 

TIME IN SECONOS 


F igure 16. Surface reverberation levels showing agree¬ 
ment with theoretical curve. Projector and receiver 
directional in vertical plane. 


On the same day (May 8, 1942) surface-reverbera¬ 
tion measurements were also carried out with the 
QCH-3 transducers oriented differently. In these ex¬ 
periments the transducers were placed at a depth of 
20 ft with the transducer axes parallel to the ocean 
surface, as before; but the long dimensions of the 
transducers were vertical instead of horizontal. With 
this transducer orientation, the correction factor 
b(6 — £,0 )b'{6 — £,0)/cos 6 cannot be neglected. The 
values of the correction factor as a function of range 
were calculated from the known directivity pattern 
of the QCH-3. A theoretical reverberation curve was 
then obtained, using equation (43) of Chapter 12, 
assuming 10 log m'/2 independent of range, and 
neglecting the term 2 A in that equation. This curve 
is plotted as the solid line in Figure 16. It can be 
compared with the points which show the actual 
reverberation levels observed in this experiment; 
each point represents the average reverberation from 
30 pings each of length 8 msec. Evidently the agree¬ 
ment between theory and experiment is quite good. 
It may be remarked that at ranges between 80 and 


Evidently if surface reverberation arises from scat¬ 
tering in a thin layer near the ocean surface, a drop 
in reverberation is to be expected when the sound 
beam is bent away from the surface, provided, of 
course, that the surface reverberation is not masked 
by volume reverberation at the range where the beam 
leaves the surface. Under conditions of sharp down¬ 
ward refraction such sudden drops have been ob¬ 
served. For example, Figures 17 and 18 show rever¬ 
beration levels obtained with the QB transducer on 
July 24, 1942. On this day ground swells were almost 
absent, but the ocean surface was scuffed up with a 
few whitecaps forming. The wind speed was 12 mph. 
The QB transducer was placed with its axis parallel 
to the surface at depths of 20 and 60 ft. Twelve con¬ 
secutive pings, each 11 msec long, were averaged at 
each depth to give the points shown in Figures 17 


b The values of JJ.fi) for both horizontal and vertical orien¬ 
tations of the QCH-3 can be obtained by using the QCH-3 
directivity patterns given in Section II of reference l, in 
equation (42) of Chapter 12. 





























































TRANSDUCER HORIZONTAL 


293 


RANGE IN YARDS 



0.01 0.05 0.1 0.5 1 5 10 

TIME IN SECONDS 


Figure 17. Surface reverberation levels with strong downward refraction. Transducer depth 20 feet. 


RANGE IN YARDS 


CD 

O 

z 

"a: 

Id 

> 

Id 

_l 

Z 

o 

H 

< 

<r 

Id 

CD 

cr 

Id 

> 

Id 

a: 


-100 


-120 


-140 


-160 


-180 






















































c 

o 

o 


>V— 

A 


















: 



\ 

\ 

% 























V 

















H 








.. . 

■v 






- O EXPERIMENTALLY observed points 

» nocFRi/m POINTS CORRECTED FOR 



\ 

V- 

— 

I 




TR 

ansducer directivity 

EORETICAL INVERSE-CUBE DEPENDENC 

.. i i i i iin 

E 


\ 

\ 




- - TH 

1 

| 



V 


< 


0.01 

TIME IN SECONDS 

Figure 18. Surface reverberation levels with strong downward refraction. Transducer depth 60 feet. 


and 18. Figure 19 shows the bathythermograph 
record and the calculated limiting rays for the two 
depths. The points marked A in Figures 17 and 18 
show the range where the limiting ray leaves the sur¬ 
face. Just as in Figure 16, the observed reverberation 
levels at short range can be expected to differ from 
a straight line with —30 log r slope because of the 
correction factor b{6 — %)b'(d — £)/cos 0. lo facili¬ 
tate comparison with this line, the observed levels 



Figure 19. Refraction conditions for data in Figures 
17 and 18. 






























































































































294 


DEEP-WATER REVERBERATION 



are corrected by increasing the experimental points 
by the value of the correction factor. That is, in 
Figures 17 and 18 the solid points are values of 


fi'ffl-MHog W-WW-W (4) 

cos e 

By using equations (41) and (43) of Chapter 12, the 
expression (4) obviously equals 


10 log y + 10 log 


(f)- 


30 log r + 10 log 


Q( 0) 


2 7T 

-2 A. (5) 


In equation (5), if A can be neglected and if m' is 
independent of range, the only dependence on range 
is contained in the term — 30 log r. Thus, with these 
assumptions the solid points in Figures 17 and 18 
should he on a straight line of —30 log r slope if re¬ 
fraction has no effect. It is seen that the first few solid 
points in Figures 17 and 18 do he on a straight line of 
—30 log r slope, but that at a range close to that when 
the limiting ray leaves the surface, there is a sharp 


drop in reverberation level. This drop does not con¬ 
tradict the theory in Chapter 12. It will be recalled 
from Section 12.3 that equation (43) is not valid and 
therefore cannot be expected to agree with measured 
levels at ranges past that at which the limiting ray 
leaves the surface. In equation (4) the correction 
factor approaches unity and its logarithm approaches 
zero as the range is increased, and in both Figures 17 
and 18 the correction is practically negligible by the 
time the sudden drop in intensity occurs. Conse¬ 
quently it is not possible to ascribe the position of the 
experimental points at ranges past the sudden drop 
in intensity to uncertainty in the value of the correc¬ 
tion factor. Thus the conclusion seems inescapable 
that the sudden drop is due to the sound rays leaving 
the surface. 

From the evidence in Figures 15 to 19, it appears 
that our basic assumption, namely that surface rever¬ 
beration arises from scattering in a thin layer near 
the ocean surface, is probably correct. The observed 
— 30 log r slope in Figure 15 and in the long-range 
portion of Figure 16 also permit the conclusion that 






























TRANSDUCER HORIZONTAL 


295 



there are times when the surface scattering coefficient 
m' is independent of the angle of incidence of the rays 
on the surface. However, it is not usually possible to 
fit the observed reverberation intensities with equa¬ 
tion (39) of Chapter 12, if m' is assumed independent 
of range. Thus, while the basic assumptions leading 
to that equation are probably correct, it cannot be 
said that the factors involved in surface reverbera¬ 
tion are completely understood. 

More illustrations of the dependence of surface 
reverberation on range may be obtained from a 
memorandum issued by UCDWR, 4 where extensive 
measurements of observed deep-water reverberation 
levels at 24 kc are summarized. This summary is 
based on data obtained on 6 cruises in the period from 
November 26, 1943 to September 1, 1944. About 110 
reverberation curves were obtained, each an average 
of five successive pings. The ping lengths used varied 
from 16 to 80 yd, but in the following curves all data 
have been corrected to the standard 80-yd length; in 
other words, the following graphs are all plotted in 
terms of the standard reverberation level. All the 
data were obtained with the JK projector at a depth 
of 16 ft on the USS Jasper with the transducer axis 
horizontal. 

Figure 20 is a plot against wind speed of all the 
reverberation levels measured at a range of 100 yd. 
All seasons of the year are represented. Thermal pat¬ 
terns were of MIKE, CHARLIE, and NAN types 
(see Chapter 5), represented respectively by dots, 
circles, and triangles. In Figure 20 a systematic in¬ 


crease in reverberation level of about 35 db is ob¬ 
served as the wind increases in velocity from zero to 
20 mph. At wind speeds of 8 mph or less, there is 
little systematic dependence on wind speed. At 
greater speeds the level rises sharply, up to speeds 
of 20 mph or more. Increase of wind speed beyond 
20 mph has little systematic effect. This dependence 
on wind speed is correlated with the roughness of the 
sea. At 8 mph the wind is strong enough to roughen 
the surface appreciably; occasionally wavelets may 
slough over, but no well-developed whitecaps are 
observed. At about 10 mph small whitecaps begin to 
appear. When the wind has reached 20 mph the sea 
is liberally covered with whitecaps. The detailed de¬ 
pendence of the appearances of the sea on wind force 
is described in a Navy manual. 5 Apparently, as the 
wind speed increases beyond 20 mph, the resulting 
increase of whitecaps causes little, if any, additional 
increase in reverberation. 

The median values of the standard reverberation 
level at 100 yd, as a function of wind speed up to 
20 mph, are roughly described by the equation 

R = -118 + 10 log (1 + 2.5 X 10-V) (6) 

where u is the wind speed in miles per hour. This 
equation is represented by the solid line in Figure 20. 
Beyond 20 mph, the function is assumed to be con¬ 
stant at R = —83 db. 

The reverberation level at long range has a 
markedly different wind-speed dependence from that 
at short range. The data in Figure 21 are taken from 
















DEEP-WATER REVERBERATION 


296 


RANGE 100 YARDS 


RANGE 1500 YARDS 


-70 


-80 


-90 


•100 


-110 


-120 


-130 




•• 




• 

•< 

• 

/ 

• 

• *'a 
r' y 

jk 

• 

»• 

: 

• 

• 

- 

/ • 
/ • 

/ 

i 

■// 
/ / 

4 / . 

<§•* 

• 



• 

; / 

/ 

, / 

•J 




• 

r 

• 





»• 




I 2 

SEA STATE 


-120 


-130 


-140 


-150 


-160 


-170 


-180 







t 

_ 

\ 

\ 

• 

• 

• ♦ 

V —v 

* ^ 

-- 

••• 

• 

• 

• < 
•• 

^**»*»-» 1 
• 7 

• 

• 

• • • 

•• 

• • 

• 

• 

• 

i 


•< 

• 














OBSERVED POINTS 
■LINE JOINING MEDIANS 


I 2 

SEA STATE 


-LINE JOINING QUARTILES 

Figure 22. Dependence of standard reverberation level on sea state. 


the same reverberation runs as the data in Figure 20, 
with the exception that the levels shown were meas¬ 
ured at 1,500 yd rather than at 100 yd. No significant 
wind-strength dependence is observed. It seems 
justifiable to conclude from these data that surface 
reverberation, which is frequently dominant at 
100 yd, has little effect at 1,500 yd; in other words, at 
1,500 yd the observed reverberation usually arises 
from volume scattering. Figure 22 shows the de¬ 
pendence of reverberation on sea state at ranges of 
100 and 1,500 yd. The 100-yd levels depend on sea 
state while the 1,500-yd levels apparently do not; 
thus the qualitative dependence in Figure 22 is the 
same as that in Figures 20 and 21. The relation be¬ 
tween wind force and sea state is given in the NDRC 
survey report on ambient noise. 6 

Figure 23 shows the reverberation levels as a func¬ 
tion of range, for high and low wind speeds. In this 
illustration are given the median reverberation levels 
and the upper and lower quartiles at each range for 
wind speeds less than 8 mph and greater than 20 
mph. It appears from Figure 23 that the reverbera¬ 
tion is entirely independent of wind speed at ranges 
greater than 1,500 yd. Actually, the quartiles at 
ranges greater than 1,500 yd are not precisely the 


same for wind speeds less than 8 mph and wind speeds 
greater than 20 mph (see Figure 5 of reference 4), but 
these differences are not thought to be significant. 
Consequently, in Figure 23, data for all wind speeds 
are used to determine the range dependence of the 
reverberation at ranges of 1,500 yd or more. 

From Figure 23, for wind speeds greater than 20 
mph, the median reverberation level drops 40 db be¬ 
tween 100 and 1,000 yd. Thus, for high wind speeds, 
the reverberation intensities usually drop off nearly 
as the fifth power of the range, rather than as the 
third power predicted in Chapter 12 on the assump¬ 
tion that 10 log m' is independent of range. (At 100 
yd or more, with the JIv at a depth of only 10 ft, the 
variation of./«(0) with range can be neglected.) Even 
faster rates of decay than the fifth power are fre¬ 
quently observed. For example, Figure 24 shows a 
plot ot the reverberation levels obtained on the Point 
Conception cruise, on March 2, 1944. The rate of 
decay in this figure is approximately as the sixth 
power of the range; that is, R decreases approxi¬ 
mately as —00 log r. The wind speed on this run was 
37 mph. 

Figure 25 shows the reverberation level at 1,500 yd 
plotted against date and area. The data fall into five 




























STANDARD REVERBERATION LEVEL R IN DB 


TRANSDUCER HORIZONTAL 


297 



IOO 200 300 500 700 1000 2000 5000 10,000 

3000 7000 

RANGE IN YARDS 

Figure 23. Dependence of standard reverberation level on range and wind speed. 



RANGE IN YAROS 

Figure 24. Sample plot of standard reverberation 
level at high wind speed. 


groups, two pairs of which were taken in the same 
area at different seasons. The data grouped in this 
way are summarized in Table 1. On the whole, Table 
1 shows that the mean reverberation levels at 1,500 
yd are independent of season and area, although the 
Cedros II and Point Conception data may indicate 
some systematic variation. 

Figure 26 is an analysis of the dependence of the 
observed reverberation levels on a parameter which 
has been found to correlate significantly with trans¬ 
mission studies at 24 kc. This parameter is the depth 
D-i in the ocean at which the temperature is 0.3 F less 
than at the surface. 7 The levels in Figure 26 are re¬ 
ferred, for convenience, to an arbitrary zero level 
which was, however, the same for all curves. Only 
data obtained after March 22, 1944 were used for 
this comparison since most of the earlier runs ended 
at about 2,000 yd. The median curves are seen to be 
practically the same for D 2 between 5 and 40 ft, 
but fall off less rapidly for D-> between 40 and 160 ft. 
Since, according to reference 7, increasing Do means 
a decreasing transmission anomaly A in equation 

































































298 


DEEP-WATER REVERBERATION 


CD 

o 

z 

cr. 


Ui 

> 

UJ 


< 

cr 

UJ 

CD 

cr 

UJ 

> 

UJ 

cr 

Q 

cr 

< 

o 

z 

< 


CEDROS I 

PT CONCEPTION 

UCDWR 

CEDROS II 

UCDWR 

NOV 1943 

MAR 1944 

MAR-APR 1944 

MAY 1944 

AUG-SEPT 1944 






• 




• 

• • 






• 



• •• 

• 




-••- 

• • 

-• MUM- 



• ••• 

-M«- 

-MM- 

• 



• 

-«*M- 





• • 

• 



AAA 










-M- 

• 

- - 



• 


-MM- 




• 

• 

• 

• 

• • • 

• 


• 

• 

• 


• • 

-M«- 

• 

• 



-••- 

• 

-— W 9 — — 


# 

-MM- 

• • 

• • •• 


• 

• 





m 







• • 




• 


- MEDIANS 





-QUARTILES 




• 






Figure 25. Standard reverberation level at 1,500 yards for various dates and areas. 


Table 1 


Area 

Month 

Median 

reverberation 

level 

at 1,500 yd 

Inter-quartile 

difference 

Number 
of cases 

All data 

Dec.-Sept. 

-140 

9 

113 

Cedros I 

November 

-141 

10 

31 

Cedros II 

May 

-146 

3 

12 

Point Conception 

March 

-137 

2 

27 

San Diego 

March-April 

-140 

6 

22 

San Diego 

August-September 

-140 

12 

21 


(26) of Chapter 12, the differences between these 
curves might be explained as a result of improved 
transmission to long ranges with larger values of D>. 
However, on the whole there is little dependence of 
the curves on the parameter Z) 2 ; certainly no de¬ 
pendence is apparent at 1,500 yards.® 

c In this connection, more recent data obtained by 
UCDWR are of interest. These data, as yet unpublished, 
show that with NAN patterns (usually falling in the class 


The results of this subsection can be summarized 
as follows. At ranges less than 1,500 yd, in standard 
24-kc echo-ranging gear oriented so that the acoustic 


5 ^ A < 10) there is frequently a hump in the reverberation 
curve at a range which corresponds to the deep scattering layer 
discussed previously. Presumably these humps are most com¬ 
mon with NAN patterns because with strong downward re¬ 
fraction the main sound beam usually strikes the deep layer 
at a well-defined range. 


































TRANSDUCER HORIZONTAL 


299 





Figure 26. Range dependence of reverberation as a 
function of refraction conditions. 


axis is parallel to the surface, the reverberation ob¬ 
served at wind speeds greater than 8 mph is pre¬ 
dominantly surface reverberation. At ranges greater 
than 1,500 yd, the reverberation does not depend 


significantly on wind speed, location, season, or ther¬ 
mal structure of the ocean. It seems justifiable, there¬ 
fore, to regard the reverberation at ranges greater 
than 1,500 yd as the characteristic volume reverbera¬ 
tion of the ocean. 

Possible Explanations 

The lack of dependence of reverberation on wind 
speed at ranges greater than 1,500 yd is well estab¬ 
lished, but is nevertheless surprising. The masking of 
the surface reverberation by volume reverberation 
at this range is in large part due to the rapid decrease 
of surface reverberation with range. The reason for 
this decrease is obscure. The following factors have 
been suggested, in Section VI of reference 1, as possi¬ 
ble causes of this rapid decrease of surface reverbera¬ 
tion with increasing range: attenuation, variation of 
the scattering coefficient with angle of incidence on 
the surface, shadowing effects of waves, and inter¬ 
ference effects in a thin surface layer (Lloyd mirror 
effect). These possible causes will now be considered 
briefly. 

In Figure 23, the drop between 100 and 1,000 yd in 
median reverberation level at wind speeds greater 
than 20 mph is 16 db more than would have been 
predicted from the —30 log r dependence of equa¬ 
tion (43), in Chapter 12. If this change is due to 
transmission loss, the term A in equation (43) must 
have a median value of 8 db per kyd. While this value 
of the attenuation is not impossible, it is significantly 
greater than the mean attenuation coefficient ob¬ 
served in transmission studies off San Diego (see 
Section 5.2.2), especially since with high wind forces 
the surface layer tends to become isothermal. If the 
steep slope in the curve of Figure 24 is due to atten¬ 
uation, the attenuation coefficient would have to be 
as high as 15 db per kyd. The possibility that some 
of the increased loss is due to attenuation cannot be 
ruled out; but on the whole the evidence from trans¬ 
mission studies does not justify regarding attenua¬ 
tion as the primary cause of the rapid decrease of 
surface reverberation with range. 

If the surface scattering layer is very thin, it can 
be argued that the scattering coefficient m' should 
decrease at least as rapidly as sin 9, where 9 is the 
grazing angle of the ray incident on the surface. For 
the total volume of surface scatterers irradiated by 
the ping at any instant is proportional to Cor, accord¬ 
ing to Section 12.3. All the energy reaching this 
volume must pass through the surface whose cross 
section in the plane of Figure 27 is AB. Thus, if I is 



















































300 


DEEP-WATER REVERBERATION 


intensity at AB, the total energy reaching the surface 
scatterers per unit time is proportional to I(AB ) = 
I (cor) sin 9. If the simple assumption is made that 
the energy scattered in all directions is the same as 
the energy scattered in the backward direction, it 
follows from the definitions of m and rn' [equations 
(4) and (32) of Chapter 12] that the total energy 


c.*-► 



OA= UPPER EDGE OF MAIN BEAM 
OB = LOWER EDGE OF MAIN BEAM 

(Figure 27. Energy reaching surface scatterers. 


scattered per unit time is proportional to m'. Thus, 
the ratio of the total energy scattered per unit time 
to the total energy reaching the scatterers per unit 
time is proportional to m '/sin 9. The former ratio can¬ 
not exceed unity; if it is to remain finite at small graz¬ 
ing angles, m' must decrease at least as rapidly as sin 9. 
If the scattering from the surface obeys Lambert’s 
law (the law of scattering of light by rough surfaces 8 ), 
then the backward scattering coefficient will be pro¬ 
portional to sin 2 9. At ranges of 100 yd or more, with 
transducers at 16 ft, sin 9 = 9 and is inversely pro¬ 
portional to the range. Thus, comparing with equa¬ 
tion (43) of Chapter 12, if the scattering arises in a 
thin surface layer, and if we can assume that equal 
amounts of energy are scattered in all directions, the 
surface reverberation would be expected to fall off as 
the fourth power of the range, or faster. Supple¬ 
mented by the added loss due to attenuation, such a 
variation of scattering coefficient with grazing angle 
could explain the observed dependence on range in 
Figures 23 and 24. 

However, before we can accept the variation of m' 
with grazing angle as an explanation of the depend¬ 
ence of surface reverberation on range, we must de¬ 
termine how thin a scattering layer is required for the 
argument of the previous paragraph to be valid. 
Figure 28 is a more exact drawing of the situation 
pictured in Figure 27, drawn so that the layer has 
appreciable thickness. In Figure 28 the projector 0 
is at depth d, and the scattering layer has thickness h. 
The scattering volume has a cross section CADE in 
the plane of the paper, with CD at long range very 
nearly equal to Cqt. Energy enters the scattering 
volume through AE (as in Figure 27) or through AC. 
From Figure 28 it is easy to obtain a simple criterion 


for the validity of the argument of the previous para¬ 
graph, if attenuation in the surface layer can be 
neglected. For, with this approximation, the energies 
entering through AC and AE are proportional re- 


; 0 t— 



spectively to the solid angles formed by rotating <t> 
and p in Figure 28 about a vertical axis (in the plane 
of the paper) through 0. At long ranges, 9 small, 
these solid angles are proportional respectively to 
angles p and p. Thus, the calculation in the previous 
paragraph of the energy entering the scattering 
volume per unit time is incorrect unless the angle < j> 
is very small compared to p. At long range we have 
approximately, with OC = r, 

h = (AC) cos 0 = r<f> cos 9 
AB = rp = (AE) tan 9 = Cqt tan 9. 

Thus the condition that p be very small compared 
to p becomes 

ft <<c (cqt) tan 9 
r cos 9 r 

For small 9, cos 9 = 1, sin 9 = 9 = d/r. Thus equa¬ 
tion (7) becomes 

ft « (Cor)-• (8) 

r 

At 1,000 yd, with 100-msec pings and d = 5 yd, 
equation (8) gives h « 30 in. There are scarcely any 
data, but it seems likely that the surface layer might 
frequently be thin enough to satisfy the relation (8). 
On the other hand, in rough seas it would not be sur¬ 
prising to find that the relation (8) is violated. 

Equation (8) was derived neglecting attenuation 
in the scattering layer; if attenuation is taken into 
account then it can be shown that the expression (8) 
must be replaced by 

1 _ e ~°rh/d <<: (coT)a> (9) 

where the attenuation in the layer, in decibels per 
yard, is 4.34a. The value of a probably depends pri¬ 
marily on the population of bubbles in the surface 














TRANSDUCER HORIZONTAL 


301 


layer and is difficult to estimate. If the attenuation 
is as large as 1 db per yd, a value which is observed 
in wake measurements (see Chapter 35), equation (9) 
would be satisfied even with very large values of h. 
In fact, it is obvious that the relation (9) will always 
be satisfied if 

(cor)a » 1. (10) 

If (8) is not satisfied, it is easy to see that (9) will not 
be satisfied if (10) is violated. 

We may summarize this discussion of the validity 
of the argument that m' should decrease at least as 
rapidly as sin 0 by stating that the argument may be 
correct, but requires further quantitative informa¬ 
tion on the thickness of the scattering layer and the 
attenuation to be expected in the layer. Until this 
information is forthcoming, the hypothesis that m' 
decreases with decreasing angle of incidence on the 
surface is not a wholly acceptable explanation of the 
variation of surface reverberation with range. 

At small grazing angles, the peaks of the water 
waves are sometimes hidden from the sound source 
by the troughs. This shadowing effect of waves also 
causes a reduction of the irradiation of the surface. 
If the scatterers are largely concentrated in a layer 
whose depth is small compared with the wave height, 
a reduction in reverberation might be expected at 
small grazing angles. Since the grazing angle decreases 
with increasing range, this effect could account for 
the rapid decrease of surface reverberation. This 
hypothesis of the shadowing effects of waves has the 
added virtue that it explains the increasing rate of 
decay with increasing sea state, since the larger the 
waves the more important this effect would be. How¬ 
ever, this hypothesis is much too qualitative to be 
accepted without further study. It can be seen that 
phenomena in high sea states may actually tend to 
make the reverberation increase with increasing 
range rather than decrease. For example, in high sea 
states, at long range, the sound rays may make large 
angles of incidence with the wave troughs, thereby 
increasing considerably the sound returned back to 
the transducer. A quantitative evaluation of the 
shadowing effect of waves is difficult and requires a 
detailed examination of surface roughness. Several 
papers issued by UCDWR 9-13 are initial attacks on 
the theory of surface scattering. That the nature of 
the surface irregularities will affect surface rever¬ 
beration seems almost intuitively obvious. Another 
report 14 describes measurements in which definite 
structure was found in surface reverberation. On this 


day there were strong swells with a wind speed of 
16 to 19 mph. Distinct blobs were observed in the 
reverberation, and these blobs altered their range at 
a rate equal to the rate at which the surface swells 
were moving. These blobs could be identified much 
more readily on the chemical recorder than on the 
oscilloscope record, where the wealth of detail con¬ 
fused the general picture. In Section VII of reference 
1, no difference was observed in reverberation meas¬ 
ured with the projector beam parallel to and perpen¬ 
dicular to the wave fronts. 

A wave in water reflected at the water-air bound¬ 
ary suffers a change in phase of the sound pressure 
(see Chapter 2). This change of phase results in inter¬ 
ference between the direct and surface-reflected rays; 
the transmission loss between the projector and 
points near the surface may be increased to a value 
much greater than the inverse square loss used in 
deriving equation (43) of Chapter 12. Furthermore, 
the increase in transmission loss will be a function 
of the range and of the distance of the scatterer from 
the surface, and will increase with decreasing depth 
and increasing range. Thus if the scatterers are lo¬ 
cated in a thin layer near the surface, this interfer¬ 
ence between direct and surface-reflected waves may 
explain the observed rapid decrease of surface rever¬ 
beration. As with the previous hypothesis of wave 
shadowing, it is necessary to make a quantitative 
investigation of this image interference effect before 
accepting it as an explanation of the range depend¬ 
ence of surface reverberation. For a plane surface, it 
is shown in Section VI of reference 1 that image inter¬ 
ference can lead to a decrease of surface reverbera¬ 
tion proportional to the seventh power of the range; 
consequently, this effect could account for the ob¬ 
served slopes of Figures 23 and 24. However, the 
surface is not plane. An approximate treatment of 
the effect of surface roughness in reference 1 shows 
that the image interference effect becomes less im¬ 
portant as the surface roughness is increased. Thus 
if image interference is causing the range dependence, 
the slope of surface reverberation should decrease 
with increasing sea state, which is contrary to what 
is observed. Another inference from the theory of the 
image interference effect is that surface reverbera¬ 
tion should increase rapidly with frequency at ranges 
where the interference is important. Unfortunately, 
there are no experiments on the variation of surface 
reverberation with frequency. 

It may be concluded from this discussion of the 
range dependence of surface reverberation that the 



302 


DEEP-WATER REVERBERATION 


reason for the rapid decrease is not understood, but 
that there are a number of factors which may play a 
part. Very likely, all the physical factors which have 
been discussed previously are included to some de¬ 
gree. In addition, there may well be other causes 
which have not been considered. It should be noted 
that at ranges less than 500 yd the measured rever¬ 
beration for wind speeds less than 8 mph decreases 
only as the inverse first power of the range in Figure 
23. This rate of decay is even slower than the pre¬ 
dicted inverse square decay of volume reverberation. 
No definite explanation has been offered for this 
feature of Figure 23, but it might be caused by a 
gradual increase in the value of the volume-scattering 
coefficient as the deep layer is approached. This effect 
would, of course, be noticeable only in sea states so 
low that volume reverberation can be measured at 
short ranges. 

14.2.2 Dependence on Ping Length 

The theoretical formulas (22), (39), and (52) for 
volume, surface, and bottom reverberation in Chap¬ 
ter 12 all have the reverberation intensity propor¬ 
tional to the ping length. The theoretical assumptions 
required to obtain this result have been discussed in 
Chapter 12. In this subsection we shall discuss 
whether or not this strict proportionality may be 
expected in practice. The only data bearing on this 
question are reported in reference 1, Section IV, and 
are summarized later. Unfortunately reference 1 does 
not state whether the reverberation studied was 
volume, surface, or bottom reverberation, but rever¬ 
beration received from ranges as low as 100 yd and 
as great as 5,000 yd was included in the analysis. 
However, ranges less than five times the ping length 
were not included. 

Figure 29 shows, qualitatively, that the reverbera¬ 
tion intensity increases with increase in the signal 
length. In that illustration, a record A of reverbera¬ 
tion following a 70-msec ping is compared with a 
record B of reverberation following a 10-msec ping. 
The attenuator settings were the same for both cases; 
however, because of the higher level of the 70-msec 
reverberation, each attenuator step was removed a 
little later for the 70-msec ping. For this reason, the 
records are directly comparable only in the intervals 
1-1, 2-2, and 3-end in which the amplification is the 
same for both records. In these intervals, the 70-msec 
reverberation is clearly much higher than the 10-msec 
reverberation. 


To test quantitatively the predicted relation 
between reverberation intensity and signal length, 
sets of data were taken on two successive days 
of the reverberation following pings of lengths 
very nearly 10, 20, 40, and 70 msec. Ten pings 
were measured on each day for each signal length. 
For each ping length, the average reverberation 
amplitude was measured at a set of logarithmically 
equispaced positions, by the band method. The 
squares of these average amplitudes were assumed 
proportional to the average reverberation inten¬ 
sities, in accordance with the usual procedure de¬ 
scribed in Chapter 13. 

The agreement between theory and experiment is 
shown in Figure 30. In that illustration, ten times the 
logarithm of the ratio of any two ping lengths is taken 
as the abscissa, and the decibel difference between 
the corresponding measured reverberation levels is 
taken as the ordinate. d If reverberation intensity 
were in fact exactly proportional to the ping length, 
all the observed points should lie on the 45-degree 
straight line drawn in the figure. In Figure 30 the 
points for which the longer ping length of a pair was 
20, 40, and 70 msec are designated differently so that 
any systematic departure depending on ping length 
can be discerned. On the whole, the agreement in 
Figure 30 between theory and experiment is satis¬ 
factory. For some reason, the agreement is better for 
the ratios involving the shorter signals (10, 20, 40 
msec) than for the ratios including the longest signal 
(70 msec). 

The deviations from the straight line in Figure 30 
are not too great to be ascribable to experimental 
error. Thus these data give no reason for doubting 
the prediction of equations (22), (39), and (52) of 
Chapter 12, that, under the conditions specified in 
that chapter, reverberation intensity should be pro¬ 
portional to ping length. However, in view of the im¬ 
portance of knowing the dependence on ping length 
for comparison of reverberation measurements made 
with different ping lengths, and for the determination 
of scattering coefficients, further investigation of this 
dependence is desirable. The measurements should 
be repeated for a wider range of ping lengths and for 
all types of reverberation. 


d The reason that the abscissas of some of the pairs of 
points in Figure 30 are not the same is that the signal lengths 
as measured from the film records were not exactly the same 
on both days. However, the equipment was set on each day 
for nominal ping lengths of 10, 20, 40, and 70 msec. 




TRANSDUCER HORIZONTAL 


303 



Figure 29. Comparison of reverberation from a 70 MS ping with that from a 10 MS ping. 


14.2.3 Unimportance of Multiple 
Scattering 

The theoretical formulas of Chapter 12 are based 
on the assumption that multiple scattering can be 
neglected. Experiments designed to measure the 
amount of multiple scattering are described in a 
memorandum by UCDWR 15 and summarized below. 

The 24-kc, QCH-3 units were mounted 6 ft apart 
with the long dimension horizontal in such a way 
that the unit used as a hydrophone could be rotated 
about a vertical axis. The unit used as a projector 
was kept in a fixed position. Observations were made 
with the receiving hydrophone at bearings of 0, 30, 
60, and 90 degrees, relative to the bearing of the pro¬ 


jector axis. That is, the receiving hydrophone was 
rotated so that it faced away from the projector, and 
the sound received in it was measured. Ping lengths 
of 15 msec were used, and 5 pings were averaged at 
each bearing. If the two QCH-3 units had been highly 
directive, interpretation of the observations would 
have been straightforward. However, they were not 
highly directive, so that the portion of the signal pro¬ 
jected in the direction in which the receiver was 
pointing could not be neglected. In order to evaluate 
the data, therefore, the following procedure was 
adopted. The expected signal in the receiving hydro¬ 
phone was calculated from the known directivity 
patterns of the hydrophone and projector, assuming 
single scattering was taking place in the ocean. It is 











304 


DEEP-WATER REVERBERATION 


easy to see from the derivation of Chapter 12 that 
this expected signal depends on the integral y*6(0,<£) 
b'(0,<t> + a)d&., where b and b' are defined as in Chap¬ 
ter 12, and a is the angle between the projector 
and hydrophone. 



0 2 4 6 8 10 12 


RATIO OF TWO PING LENGTHS IN DB 

Figure 30. Observed dependence of reverberation in¬ 
tensity on ping length. 

The values of the above integral were computed 
for a equal to 0, 30, 60, and 90 degrees, and the pre¬ 
dicted reverberation intensities were then compared 
with the observed intensities. It was found that the 
calculated levels were within 1 to 2 db of the observed 
average levels for ranges up to about 200 yd, beyond 
which no measurements were made. Since the experi¬ 
mental error of the measurements was not less than 
1 to 2 db, these results show that at short ranges 
multiple scattering makes a negligible contribution 
to the received reverberation. However, these results 
give no information about the effect of multiple 
scattering at longer ranges. 

It is easy to show that multiple scattering can 
certainly be neglected if the volume scattering in all 
directions is the same as in the backward direction. 
For, in this event, the total energy scattered per 
second per unit intensity at dV is just mdV (see 
Section 12.1). The loss in intensity dl in a distance dx 
of a plane wave of intensity I traveling in the x 
direction is then 

dl = mldx 
which gives I = 7 0 e -mx . 

Thus under these circumstances the attenuation of a 


sound wave by scattering is 4.34 X 10 3 m db per kyd. 
It is shown later that m is rarely greater than 10 -5 . 
By using this value of m, the attenuation due to 
scattering is 4.34 X 10 -2 db per kyd. Now, with any 
kind of a transducer, but especially with a direc¬ 
tional transducer, multiple scattering will not be im¬ 
portant in reverberation until the amount of singly 
scattered energy in the ocean becomes appreciable 
compared to the amount of energy remaining in the 
direct sound beam. Obviously, with a scattered 
energy loss of only 4.34 X 10 -2 db per kyd, scattered 
energy in the ocean is negligible compared to the 
energy in the direct beam for ranges less than 
20,000 yd where the total scattered energy loss is 
not yet 1 db. 

Despite the arguments of the preceding paragraph, 
more experimental evidence bearing on multiple 
scattering would be desirable especially since the 
oblique scattering may be appreciably different from 
the backward scattering. One way to check the im¬ 
portance of multiple scattering would be to compare 
with experiment at long ranges the predicted de¬ 
pendence of received reverberation intensity on trans¬ 
ducer directivity. If multiple scattering is important, 
the difference between reverberation levels measured 
with directional and nondirectional transducers will 
not be given by J v [equation (21) of Chapter 12]. No 
such measurements have been reported; in fact, the 
whole question of comparing with experiment the 
dependence of reverberation on the theoretical rever¬ 
beration indices J„ and J s seems to have been neg¬ 
lected. Knowledge of this dependence is required for 
comparison of measurements made with different 
gear, and also for prediction of the effect on reverber¬ 
ation in echo-ranging gear of changes in gear direc¬ 
tivity. There is no reason for doubting the validity 
of the formulas of Chapter 12 for ordinary gear, but 
with highly directive gear, multiple scattering and 
other effects may produce deviations from the theo¬ 
retical formulas. A UCDWR internal report 16 de¬ 
scribes experiments in which the measured vertical 
directivity patterns in the ocean were very different 
from the patterns obtained at a calibrating station. 
Pitch and roll of the echo-ranging vessel will also 
cause deviations from the predicted reverberation in¬ 
tensities, especially for surface reverberation. 17 

14.2.4 Average Levels 

Figure 31 is a summary of the measured 24-kc re¬ 
verberation levels reported in reference 4. The levels 
shown are standard reverberation levels, as defined 





























TRANSDUCER HORIZONTAL 


305 



by equation (25) of Chapter 12. The dots show the 
lowest reported reverberation levels at each range, 
and the circles the highest levels. Median values of 
the data are shown by triangles. At ranges less than 
1,500 yd, the lower triangles are the median values 
for wind speeds less than or equal to 8 mph, and the 
upper triangles are the median values for wind speeds 
greater than or equal to 20 mph. 

These data are neither an adequate nor a random 
sample, and detailed analysis of them is not justi¬ 
fiable. However, some interesting inferences, which 
should be reasonably reliable, can be drawn from the 
data. The lower solid line in Figure 31 is a plot of 
equation (26) of Chapter 12 against range with 
10 log m set equal to —80 db, A set equal to 3 db 
per kyd, and Ai set equal to zero. The upper solid line 
is drawn with 10 log m set equal to —60 db, with A 
set equal to 1.5 db per kyd, and Ai set equal to zero. 
Both lines were plotted with J v set equal to —25 db. 18 
If all reverberation at ranges greater than or equal 
to 1,500 yd is volume reverberation, and if the lower 
levels at shorter ranges are volume reverberation, 
then the upper and lower solid curves would represent 
estimated upper and lower limits to 24-kc volume- 
reverberation levels. 

The middle solid curve is drawn with 10 log m set 
equal to — 60 db, and with A set equal to 4 db per 
kyd. This curve fits the median values of reverbera¬ 


tion for wind speeds less than or equal to 8 mph 
surprisingly well at ranges from 500 to 3,500 yd; it 
can probably be assumed that these median values 
for wind speeds less than or equal to 8 mph represent 
volume reverberation. The 4 db per kyd value of A 
is gratifyingly close to the average value measured in 
transmission studies under good conditions (see Sec¬ 
tion 5.2.2). These results apparently indicate that a 
scattering coefficient 10 log m equal to — 60 db is a 
typical value for the volume reverberation from 
horizontally projected 24-kc sound beams. Oc¬ 
casionally, however, the scattering coefficient be¬ 
comes very small, 10 log m becoming as low as 
-80 db. 

It does not seem legitimate to attempt any con¬ 
clusions based on these differences in the values of A 
required to fit the curves of Figure 31. Both A and m 
are highly variable quantities, and are known to be a 
function of depth in the ocean. However, comparison 
of the upper and median curves at ranges past 
1,500 yd suggests that differences in the long-range 
reverberation levels may be frequently due merely to 
variations in A. 

It has already been pointed out that the deep 
scattering layers discussed in the first portion of this 
chapter will tend to increase the reverberation at 
long range above the levels otherwise expected. 4 
Studies of bottom reverberation (see Chapter 15) 














































306 


DEEP-WATER REVERBERATION 


have shown that the main beam from standard 24-kc 
gear will usually reach a depth of 1,000 ft at a range 
of about 2,000 yd. The median curve for volume re¬ 
verberation in Figure 31 does not show any evidence 
of an increase in the volume-scattering coefficient at 
long ranges; it will be recalled that the value of 
10 log m in these deep layers frequently exceeded the 
mean value of 10 log to in the ocean by 15 db or more. 
However, it appears that this failure to observe the 
deep layer must have been due to sampling. More 
recent studies by UCDWR, still unpublished, show 
that the deep layer is frequently discernible as a very 
definite bulge on the reverberation curve. These new 
data also show that the maximum value of 10 log to 
is — 50 db or perhaps even slightly higher rather than 
the —60 db value indicated by Figure 31. 

From Figure 31 and from equation (26) of Chap¬ 
ter 12, we may conclude that the backward volume- 
scattering coefficient to for horizontally projected 
24-kc beams varies between 10~ 6 and 10~ 8 per yd 
with 10 -6 the typical value. It will be noted that the 
dimensions of to in equation (26) of Chapter 12 are 
per yard; values of to expressed per foot will be one- 
third or 5 db less. Also, from Chapter 12, we recall 
that this value of to is about 3 db greater than the 
“true” value of the backward-scattering coefficient 
of the ocean. Since equation (45) of Chapter 12 does 
not describe the range dependence of surface rever¬ 
beration very well, determination of the surface 
scattering coefficient m' from comparison of that 
equation with Figure 31 is not very meaningful. 
However, if we make the comparison, with A set 
equal to 4 db per kyd and J, set equal to —16 db, 18 
the median values of 10 log to' at ranges of 100 and 
1,000 yd for wind speeds greater than 20 mph are 
respectively —22 db and —31 db. (Note that to' is a 
dimensionless quantity.) 

It will be recalled that at the beginning of this 
chapter we assumed that surface reverberation could 
be eliminated by pointing the transducer downward. 
Off the main lobe, the response b(Q,4>) of standard 
24-kc transducers is usually assumed to average 
about 30 db less than the peak response on the main 
lobe, 19 but this is only an approximate average value. 
Using this 30-db estimate, we see from Figure 31 that 
at ranges of 100 yd, in high sea states, surface rever¬ 
beration may exceed volume reverberation even with 
the transducer directed downward, but only if the 
volume reverberation level is close to the minimum 
values observed. At ranges greater than about 500 
yd, or with wind speeds less than about 15 mph, 


pointing the transducer downward should usually 
eliminate surface reverberation (see Figure 20). This 
estimate of the wind speeds and ranges at which sur¬ 
face reverberation can be eliminated by pointing the 
transducer downward assumes, however, that the 
volume reverberation with the transducer pointed 
downward is the same as when the transducer is hori¬ 
zontal. Measurements reported in reference 2 sug¬ 
gest that the volume scatterers are anisotropic and, 
specifically, have a smaller backward-scattering 
coefficient when the sound arrives from a vertical 
direction than when the sound arrives from a hori¬ 
zontal direction. However, the observed difference in 
10 log to was only about 6 db and thus hardly affects 
the conclusion stated previously that surface rever¬ 
beration can almost always be eliminated by pointing 
the transducers downward. 

i t.2.5 Scattering Coefficient of a 
Layer of Bubbles 

It is of interest to compare the median values of 
10 log in' obtained from Figure 31 with the values ex¬ 
pected if the surface consisted of a dense layer of 
resonant bubbles. The theory of air bubbles in water 
is given in Chapter 28; the geometry of scattering by 



Figure 32. Scattering from surface layer of bubbles. 

such a layer is illustrated in Figure 32. By definition, 
a densely populated layer of bubbles is one in which 
the attenuation is so high that there is essentially 
infinite transmission loss through the layer. For this 
reason, in Figure 32, energy reaches the scatterer at 
X and returns to the scatterer at O along the direct 
path OAX only; scattering along a path reflected 
from the air-water interface, such as OBX, can be 
neglected since almost no energy reaches B. It fol¬ 
lows from bubble theory that multiple scattering can 
be neglected as well. Neglecting refraction, the 
expected reverberation intensity can now be calcu¬ 
lated directly from equation (29) of Chapter 12, 
using 

h = e —e~ Na ' D (11) 

r- 

m = No, (12) 




TRANSDUCER HORIZONTAL 


307 


where N is the number of bubbles per cubic centi¬ 
meter, <j e and a s are respectively the absorption and 
scattering cross sections of a resonant bubble (de¬ 
fined in Chapter 28), D is the distance AX, a the 
usual attenuation coefficient equal to about 4 db per 
kyd, and r the range. 

Evaluating the integral in equation (29) of Chap¬ 
ter 12 with the aid of equations (11) and (12), 
and, comparing the result with equation (39) of 
Chapter 12, it is readily found that 


where 0 is as usual the angle of elevation of the ray 
from the projector to the scatterer at range r (Figure 
7, Chapter 12), and d the projector depth. For a trans¬ 
ducer at 16 ft, equation (15) gives m' equal to —28 db 
at 100 yd and —38 db at 1,000 yd (using Figure 1 
in Chapter 34). These values are about 6 db lower 


than the median measured values of —22 db and 
— 31 db and are still lower than the highest reported 
levels at 100 and 1,000 yd (see Figure 31). Since equa¬ 
tion (13), derived on the basis of a densely populated 
surface layer, gives the highest possible value of m' 
for scattering by bubbles, it seems that measured 
surface reverberation levels cannot be explained on 
the hypothesis of scattering by a surface layer of 
bubbles. It will be noted that, because of the assumed 
neglect of scattering along OBX (Figure 32), in 
this situation of scattering by a densely popu¬ 
lated surface layer of bubbles the value of m! indi¬ 
cated by equation (13) is the true surface-scattering 
coefficient. Thus, although the argument presented 
in Section 12.5.6, that measured values of m' are 6 db 
greater than the true value of the surface-scattering 
coefficient, is probably valid, it is evident that the 
validity of this 6-db relation depends on the physical 
process which gives rise to the surface reverberation. 



Chapter 15 


SHALLOW-WATER REVERBERATION 



PROJECTOR DOWN 90 DEGREES 



PROJECTOR DOWN 30 DEGREES 



PROJECTOR UP 30 DEGREES 



PROJECTOR HORIZONTAL 

Figure 1. Expected behavior of bottom reverberation 
for idealized projector. 


C omparison of theory and experiment in bottom 
reverberation is complicated by the directivity 
pattern of the transducer and by uncertainty regard¬ 
ing the dependence of the scattering coefficient on the 
angle of incidence. An additional complication is 
refraction, which is of considerable importance in de¬ 
termining bottom-reverberation levels. One way in 
which bottom reverberation differs from the other 
types of reverberation we have considered is that 
bottom reverberation is not heard immediately after 
the initial ping, but appears some time later, usually 
coming in as a distinct crash. This delay results from 
the fact that the bottom, unlike the scatterers re¬ 
sponsible for surface and volume reverberation, is 
usually a significant distance from the projector. 

15.1 QUALITATIVE DESCRIPTION OF 
BOTTOM REVERBERATION 

Figure 1 illustrates the expected behavior of the 
bottom reverberation for an idealized projector hav¬ 
ing constant sound output within 5 degrees of the 
axis and zero output outside 5 degrees. For illustra¬ 
tive purposes, we may assume there is no refraction. 
Then for this simple type of sound beam, scattered 
sound is received, at the time instant t, only from 
those scatterers included within a sector of a spherical 
shell centered at the projector and bounded by the 
limits of the sound beam. The mean radius of this 
shell is ct/2 and its thickness cr/2, as pointed out in 
Section 12.2. Bottom reverberation is received when¬ 
ever this shell cuts off some portion of the bottom. 

Bottom reverberation will set in at the time cor¬ 
responding to the shortest range at which the beam 
strikes the bottom. Since bottom scattering coeffi¬ 
cients are usually relatively large, the total received 
reverberation will increase sharply at the time of on¬ 
set of bottom reverberation. Scattering at the bottom 
will cease, except for sound which is reflected or scat¬ 
tered toward the bottom, at the time the last portion 
of the beam leaves the bottom. 

The case of vertical incidence on the bottom is 
illustrated in the first box of Figure 1. In this case, as 
shown in the box, all portions of the beam strike the 


308 



























QUALITATIVE DESCRIPTION OF BOTTOM REVERBERATION 


309 


bottom nearly simultaneously. Thus the reverbera¬ 
tion begins and ends very abruptly. The time of on¬ 
set of the reverberation equals 2 d/c, where d is the 
depth of the bottom below the projector, and c is the 

SURFACE 



Figure 2. Vertical incidence of ping on bottom. 

sound velocity. Evidently, as shown in Figure 2, all 
portions of the beam do not strike the bottom exactly 
at the same time so that the duration of the rever¬ 
beration does depend somewhat on the beam width. 
For narrow beams, it is easily shown that the rever¬ 
beration duration is given by 

Reverberation duration = r + —, (1) 

4c 

where a is the beam width, shown in Figure 2, and r 
the ping duration. Thus, with very short pings inci¬ 
dent vertically on the bottom, the duration of the 
reverberation may appreciably exceed the ping dura¬ 
tion. 

When the beam is incident on the bottom at some 
slant angle, all parts of the beam do not strike the 
bottom at the same time. Consequently the rever¬ 
beration does not begin or end quite as sharply 
as in the previous case, and the reverberation 
duration is greater than the ping duration. This 
case is illustrated in the second box of Figure 1. 
In this situation it is easily shown from Figure 3 that 
the time of onset and duration of the reverberation 
are given by 



Reverberation duration = 

t + ~ csc (* + cot (^ _ i)' ^ 

If the beam is pointed up toward the surface, or is 
horizontal, surface reverberation will be heard before 


the bottom reverberation begins to come in. With a 
horizontal beam, different parts of the beam strike 
the bottom at widely spaced intervals, so that the 
reverberation lasts a long time. These statements are 
illustrated in the third and fourth boxes of Figure 1. 

Of course, the sound beam is not actually confined 
within a cone; some sound is sent in all directions. 
However, most sound projectors commonly in use 
confine all but a small fraction of the emitted and 
received sound within small angles from the beam 
axis, so that the simple description of Figure 1 should 
be a good approximation to the observed phenomena. 
Figure 4 is an experimental illustration of the qualita¬ 
tive predictions of Figure 1. The data making up 
Figure 4 were obtained in an area where the water 
depth was 72 ft. It is noticed that the reverberation 
recorded with the projector directed vertically down¬ 
ward consisted of a single pulse of about the duration 
of the ping. The reverberation levels recorded with 
the projector directed 30 degrees down fall on a curve 
approximating the simple case shown in the second 
box of Figure 1. There is a rapid rise of reverberation, 
due to bottom scattering, at the time corresponding 
approximately to the range at which the main beam 
first strikes the bottom (easily calculated as 35 yd for 
a half beam width of 6 degrees) and reaching a peak 
at about the range where the axis of the beam arrives 


SURFACE 



Figure 3. Slanting incidence of ping on bottom. 

at the bottom (40 yd). The peak is followed by a 
rapid decay. The case of the horizontal beam is il¬ 
lustrated in the bottom curve of Figure 4, and is seen 
to correspond roughly to the bottom curve in Figure 
1. The initial reverberation recorded at 40 to 50 yd is 
received from the bottom on the projector side lobe; 
this type of reverberation decays rapidly for about 









310 


SHALLOW-WATER REVERBERATION 











- 


o 





BEAM DOWN 90 DEGREES 

PING LENGTH s 4 YDS 




























< 




40 80 120 160 200 240 280 320 360 


RANGE IN YARDS 


O MEASURED LEVELS 


- PREDICTED LEVELS 


Figure 4. Observed and predicted bottom reverberation levels. 


0.01 sec; then there is a rather rapid growth in the 
reverberation intensity when the main beam reaches 
the bottom. Further illustrations of the reverbera¬ 
tion measured with the beam down 30 degrees are 
shown in Figure 5. The ocean depth was 48 ft and 
the projector depth 10 ft so that the peak of the 
reverberation is observed to come in at a time cor¬ 
responding to a range of about 25 yd. The ensuing 
lesser maxima are the result of successive multiple 
reflections from surface and bottom. These observa¬ 
tions were taken over a bottom thickly covered with 
boulders of the order of one foot in diameter. An 
interesting feature of these curves is the change in 
the duration of the main reverberation pulse as the 
frequency changes. The duration of the pulse de¬ 
creases progressively with increasing frequency; this 
effect is due to decrease in beam width. In the case of 
the projector directed upward at an angle of 30 de¬ 
grees, it is seen from Figure 5 that the surface rever¬ 
beration peaks predicted in Figure 1 are lacking. This 
absence is due to the fact that the reverberation was 


measured under conditions which combined very 
shallow water with a smooth sea surface and a rough 
sea bottom, so that bottom reverberation masked 
surface reverberation at practically all ranges. Under 
other circumstances, with a smooth bottom and a 
rough sea, or deeper water, the peak of surface rever¬ 
beration can usually be observed before the crash of 
bottom reverberation comes in. 

These remarks indicate that bottom reverberation, 
under at least some circumstances, behaves quali¬ 
tatively as would be expected from a simple geomet¬ 
rical analysis of the time required for the sound to 
reach the bottom. The next step in the analysis is to 
attempt to make a quantitative prediction of the 
expected reverberation levels, and then to compare 
these theoretical predictions with experiments. 

From formula (54) of Chapter 12, it is clear that 
the received reverberation depends on the transmis¬ 
sion loss to and from the bottom, on the transducer 
directivity, and on the scattering strength of the 
bottom. It can be assumed that the transducer 
























































REVERBERATION LEVEL IN DECIBELS 


QUALITATIVE DESCRIPTION OF BOTTOM REVERBERATION 


311 


BEAM DOWN 30 DEGREES 
PROJECTOR DEPTH * 10 FT 


BEAM UP 30 DEGREES 
PROJECTOR DEPTH = 38 FT 












i 

80 KC 

T : a Ytnnt; 




A 







1 








I 


\ 


\ 



0 32 64 96 128 0 32 64 96 128 


RANGE IN YARDS 

Figure 5. Observed reverberation level with beam inclined 30 degrees. 




















































































































































312 


SHALLOW-WATER REVERBERATION 


directivity is known in sufficient detail, although in 
practice it may prove difficult to obtain even this 
information. The scattering strength of the bottom 
is in general a function of the angle of incidence of the 
rays striking the bottom. Temperature gradients in 
the ocean are frequently sufficiently large that the 
sound rays in the main transducer beam have suf¬ 
fered appreciable bending by the time they reach the 
bottom. For this reason bottom reverberation is 
likely to depend much more strongly on transmission 
conditions than surface reverberation. To accurately 
predict bottom reverberation levels, therefore, it is 
necessary to have detailed knowledge of the ray paths 
and transmission loss to the bottom. 

Unfortunately, there are practically no reported 
bottom reverberation measurements for which the 
transmission to the bottom can be regarded as known 
in detail (including knowledge of the ray paths and 
the transmission loss along these paths). Conse¬ 
quently, any comparison of predicted reverberation 
levels with experimental observations must be based 
on assumptions about the transmission; and detailed 
agreement in any single experiment between pre¬ 
dicted bottom-reverberation levels and observed 
levels should not be expected. Rather, because of this 
uncertainty concerning the transmission from the 
transducer to the bottom, the comparison of theory 
and experiment becomes even more a purely statis¬ 
tical process than was the case for surface and volume 
reverberation. 

This statistical approach, which is described later, 
is in some respects justifiable. Transmission studies 
made to date reveal little likelihood that detailed 
knowledge of the transmission can be obtained 
aboard an ordinary echo-ranging warship in any 
practicable way. What is required is a statement of 
the average reverberation levels to be expected for 
various broad classifications of echo-ranging gear, 
transmission conditions, and bottom types. 

15.2 EFFECTS OF REFRACTION 

Some of the results of a statistical analysis of bot¬ 
tom reverberation are described in an internal report 
by UCDWR. 1 In this report, many bottom rever¬ 
beration curves were plotted against range. The data 
comprising these curves were all taken at 24 kc with 
standard Navy gear directed horizontally, but were 
obtained in a variety of regions, over many different 
types of bottoms, at differing depths, and with vary¬ 
ing refraction conditions. Examination of these data 


showed a number of similar features on almost all the 
curves. In general, the curves showed the following 
characteristics. 

1. A peak which comes in shortly after the out¬ 
going signal and results from surface reverberation. 

2. A rapid decay of reverberation with the level 
reaching a minimum at a range of two times the 
depth of water. 

3. A broad rise in level as the range increases, de¬ 
veloping a second peak at a range of about six times 
the depth of water. This rise is due to bottom rever¬ 
beration. 

4. Beyond the second peak a rapid decrease of in¬ 
tensity, approximately proportional to the inverse 
fourth power of the range. However, very large varia¬ 
tions from this type of decay were observed. 

The range of the bottom reverberation peak de¬ 
pends of course on refraction; for this reason, the 
dependence of the range of this peak on depth be¬ 
comes a statistical problem. If there were no refrac¬ 
tion, in other words, if the sound rays always traveled 
in straight lines, then the ratio of the range of the 
peak to the depth (over plane and smooth bottoms) 
would obviously always be a fixed quantity depend¬ 
ing only on the directivity pattern of the transducer. 
In fact, for standard 24-kc echo-ranging gear, with 
a beam width of 5 to 6 degrees, the range of the peak 
would be 10 to 12 times the water depth, if refraction 
were absent. 

In order to judge the usefulness of the statistical 
study in reference 1, it is necessary to know what 
kinds of temperature gradients were included, and 
the extent to which these refraction patterns obtained 
near San Diego are typical of refraction conditions 
in other localities. There are reasons for believing that 
the results of reference 1 may be valid for a wide 
variety of temperature gradients. Bathythermograph 
patterns are not completely arbitrary in shape; posi¬ 
tive gradients are relatively rare, with the result that 
most patterns other than isothermal ones show a 
continuous decrease of temperature between surface 
and bottom. Furthermore, the effect of refraction is 
greatest for horizontal or nearly horizontal rays. 
Once the rays have been bent through an appreciable 
angle, the amount of additional bending, even by 
quite sharp negative gradients, is relatively small. 
For these reasons, the ratio of the range of the peak 
to the depth may be expected to be relatively con¬ 
stant for a wide variety of gradients excluding isother¬ 
mal or “nearly isothermal” types, where, for the 
purpose of this discussion, “nearly isothermal” water 



EFFECTS OF REFRACTION 


313 



2 5 10 20 30 40 50 60 70 80 90 95 98 99 

PERCENT GREATER THAN INDICATED VALUE 

Figure 6. Cumulative distribution of observed ratio: 

Range to reverberation peak 
Water depth 


may be defined as water in which the temperature 
at the bottom differs from the temperature at the 
surface by less than five degrees. 

Actually, examination of the data in reference 1 
shows that relatively few of the reverberation curves 
analyzed were obtained in isothermal or nearly iso¬ 
thermal water. Thus, the results of that study do not 
apply to water in which the top-to-bottom tempera¬ 
ture change is less than five degrees. This fact helps 
to account for the disparity between the observed 
range of the reverberation peak, characteristically 
about six times the depth, and the predicted value 
of 10 to 12 times the depth for isothermal water. 

After these preliminary remarks, we may examine 
the San Diego results in more detail. Figure 6 is a 
cumulative plot, taken from reference 1, of the ratio 
of the range of the bottom reverberation peak to the 
water depth. The median point on this curve cor¬ 
responds to a range-depth ratio of 6.2. Fifty per cent 
of all peak ranges were found to lie between 5.1 and 
7.2 times the depth, and 80 per cent to lie between 
4 and 8 times the depth. These results agree well 
enough with the results of another study by UCD YVR. 2 
In reference 2, which, however, was based on a 
smaller number of reverberation curves, the average 
range to the peak was about 5 times the depth. The 
difference between these two estimates of the ratio 
of the range of the peak to the water depth is prob¬ 
ably due to sampling and to the fact that the rever¬ 
beration curves plot only a few isolated points of the 
measured film. The data discussed in reference 2 are 
described in somewhat more detail in Section 15.3.1; 


they were obtained by using a transducer whose 
beam pattern was similar to that of standard Navy 
echo-ranging gear. Bathythermograph data and ray 
diagrams were available, and it was found that the 
range to the reverberation peak corresponded to 
about the range where the 6-degree ray reached the 
bottom. In another internal report by UCDWR, 3 it 
was found that the range of the peak usually cor¬ 
responded to the range at which the 5-degree ray 
reached the bottom. 

For standard Navy gear at 24 kc, the half beam 
width y defined in Figure 4 of Chapter 12 is close to 
6 degrees. Thus, the results described in the preceding 
paragraph suggest that with standard gear at 24 kc 
the range where the beam’s edge (5- or 6-degree ray) 
strikes the bottom is the range of the reverberation 
peak. These results suggest furthermore that in water 
which is not “nearly isothermal” the range of the 
reverberation peak is between four and eight times 
the depth. For simple temperature gradients, it is 
easy to estimate the range at which various rays 
will strike the bottom, as a function of the depth to 
the bottom. 4 Table 1 gives the results of such calcu¬ 
lations, for various initial ray angles, water depths, 
and surface-to-bottom temperature differences; in 
computing this table it was assumed that the tem¬ 
perature decreased linearly from the surface to the 
bottom. It is clear from the table that with linear 
gradients the 6-clegree ray always does strike the 
bottom at a range between 4 and 8 times the water 
depth, when the temperature difference between the 
projector and bottom is greater than 5 degrees. 





















314 


SHALLOW-WATER REVERBERATION 


Table 1. Ranges at which rays leaving the projector at different angles strike the bottom. 


Temperature 
difference 
between 
projector 
and bottom 
in degrees 

Depth 
between 
projector 
and bottom 
in feet 

Range at which ray strikes bottom 
in yards 

Ratio of 
range at which 
6-degree ray 
strikes bottom 
to water depth 

Angle of ray leaving projector 
in degrees 

4 

6 

8 

12 

5 

50 

171 

132 

106 

74 

7.9 


100 

345 

264 

211 

148 

7.9 


200 

696 

534 

425 

299 

8.0 


300 

1070 

809 

640 

449 

8.1 

10 

50 

141 

115 

96 

70 

6.9 


100 

282 

230 

194 

141 

6.9 


200 

571 

464 

385 

282 

7.0 


300 

863 

700 

580 

425 

7.0 

15 

50 

123 

103 

88 

67 

6.2 


100 

244 

207 

176 

133 

6.2 


200 

491 

415 

356 

268 

6.2 


300 

743 

628 

534 

402 

6.3 

20 

50 

109 

94 

82 

63 

5.6 


100 

219 

188 

162 

127 

5.6 


200 

440 

373 

329 

253 

5.7 


300 

664 

568 

496 

383 

5.7 


15.3 BOTTOM SCATTERING COEFFICIENTS 

Having established the probable limits of the 
range to the reverberation peak, it is desirable to 
estimate the height of the reverberation peak. This 
height has been found to depend markedly on the 
type of bottom. In general, reverberation is highest 
over ROCK, less over SAND-AND-MUD or MUD, 
and least over SAND, although in some cases rever¬ 
beration over SAND has been reported to be quite 
high, especially after a storm when rippling of the 
bottom may be the cause. 1 The relative values of 
the reverberation over these bottoms may be ex¬ 
pressed in terms of the bottom-scattering coefficient 
m", by using equation (54) of Chapter 12 if the trans¬ 
ducer directivity and the transmission to the bottom 
are known in detail. Detailed information concerning 
directivity and transmission has not usually been 
available, but it has proved possible to determine 
average values for the bottom scattering coefficients 
by making reasonable assumptions about the trans¬ 
mission. 

In this section, we shall summarize present infor¬ 
mation concerning the variation of the bottom scat¬ 
tering coefficient m". The three factors which are ex¬ 
pected to be most important in determining the value 
of m" are (1) the grazing angle of the sound incident 


on the bottom, (2) the frequency of the incident 
sound, and (3) the nature of the bottom. These 
factors will be considered in the same order. 

15.3.1 Dependence on Grazing Angle 

Knowledge of the nature of the dependence of m" 
on grazing angle is necessary for the detailed predic¬ 
tion of bottom reverberation as a function of range; 
in addition, the nature of this dependence is of 
theoretical interest. It has been shown in Chapter 14 
that, with simple assumptions about the law of 
scattering, for a very thin scattering layer the value 
of the backward scattering coefficient should de¬ 
crease with decreasing grazing angle at least as 
rapidly as sin and for scattering obeying Lambert’s 
law the backward scattering coefficient should be 
proportional to sin 2 d. Thus, determination of the 
dependence of m" on angle can furnish information 
about the law of scattering at the bottom and can 
also be used to check the validity of our ideas about 
bottom reverberation. 

However, determination of this dependence is not 
easy. It would appear at first thought that the de¬ 
pendence would be an easy by-product of the analysis 
of ordinary reverberation runs with horizontal trans¬ 
ducers if temperature-depth data were also available. 


















BOTTOM SCATTERING COEFFICIENTS 


315 


CD 

O 


cr . 

_i 

UJ 

> 

UJ 


1 

<£ 

UJ 

CD 

<r 

UJ 

> 

UJ 

a: 




0.1 

0.2 0.5 1 

0.1 

0.2 0.5 1 

2 


TIME IN SECONDS 


TIME IN SECONDS 



COARSE SAND DEPTH 170 FEET 

AUGUST 18, 1943 


SAND-AND-MUD DEPTH 315 FEET 
SEPTEMBER 27, 1943 



RANGE IN YARDS RANGE IN YARDS 



0.1 0.2 0.5 I 


TIME IN SECONOS 

MUD DEPTH 255 FEET 
OCTOBER 9,1943 



0.1 0.2 0.5 I 2 

TIME IN SECONDS 

ROCK OEPTH 370 FEET 
SEPTEMBER 27, 1943 


I CALCULATED RANGE AT WHICH 
T 6° RAY STRIKES BOTTOM 

Figure 7. Typical bottom reverberation levels with horizontal transducer. 


The reverberation level at any range could be trans¬ 
lated into the bottom scattering coefficient by use of 
equation (54) of Chapter 12, and the grazing angle of 
the sound at this range could be computed from the 
temperature-depth information. However, it will be 
seen in the following subsection that this procedure 
is not workable, briefly because the value of m" com¬ 
puted in this way is accurate only for the portion of 
the bottom struck by the central portion of the pro¬ 
jected beam, and within this limited area there is not 
much variation in the grazing angle. Thus, in order to 
determine the form of the function specifically 

designed experiments are necessary. 


An Experiment Designed to Measure m”(9) 

Data casting light on the angular dependence of m" 
were obtained in a series of experiments, made in 
August, September, and October of 1943, and de¬ 
scribed in an internal report by UCDWR. 2 On each 
day that measurements were taken, the transducer 
was set either at 0 degrees (main transducer beam 
horizontal) or at 30 degrees (main transducer beam 
pointed 30 degrees down from the horizontal), and 
lowered to a depth of 9 ft. Bottom reverberation for 
a number of 10-msec 24-kc pings was then recorded 
as a function of time, by using the equipment de¬ 
scribed as D in Section 13.1.1. By comparing measure- 






































































316 


SHALLOW-'WATER REVERBERATION 


ments made with the two different transducer orien¬ 
tations in the same or similar areas, it should be pos¬ 
sible to obtain some information about the angular 
dependence of m". The following paragraphs describe 
briefly the analysis of these data made in reference 2; 
it is convenient to begin by considering the 0-degree 
data. 

For the purpose of analyzing the 0-degree data, the 
reverberation records obtained were segregated into 
nineteen groups, each group comprising at least nine 
records of bottom reverberation taken at nearly the 
same time on a single day over one of six bottom 
areas. The records were then measured and averaged 
over the group to give the mean reverberation ampli¬ 
tude, and the average reverberation level was plotted 
against range for each group. Typical curves ob¬ 
tained are shown in Figure 7. These curves extend 
only to the range at which the reverberation becomes 
comparable to the recording background. On each 
curve in Figure 7 is shown the range at which the 
6-degree ray struck the bottom, as computed from 
the measured BT pattern. The high levels of the first 
plotted points at 100-yd range in the curves of 
Figure 7 are due to surface reverberation. 

These reverberation curves for horizontal trans¬ 
ducers were then compared with equation (54) of 
Chapter 12, by using 4 db per kvd for the absorption; 
in addition, the anomaly due to refraction was com¬ 
puted from the ray diagram drawn from the BT 
pattern, according to the methods described in Chap¬ 
ter 3. The total correction for the anomaly was 
found to be small for ranges corresponding to the re¬ 
verberation peak. The average magnitude of twice the 
anomaly correction for those ranges was only 2.5 db; 
but the uncertainty in the transmission anomaly led 
to uncertain values of m" corresponding to the 
reverberation at longer ranges. Nevertheless, by us¬ 
ing the data and comparing with equation (54) of 
Chapter 12, it was possible in this way to obtain for 
each group of records a curve for m", the bottom 
scattering coefficient, as a function of the range of 
the reverberation. At each range the incident grazing 
angle of the ray reaching the bottom was computed 
from the refraction diagram. However, from these 
curves of m" against range, as explained in more de¬ 
tail later, the value of m" was accurately determi¬ 
nable only for grazing angles on the bottom very 
nearly equal to the grazing angle of the central ray 
of the main beam. This angle, for all the horizontal 
projector curves studied in reference 2, lay between 
9 and 13 degrees. It is clear therefore that determina¬ 


tion of the angular dependence of m" was not possible 
from the 0-degree data alone. 

By using the 30-degree data, however, the value of 
m" when the grazing angle on the bottom is equal to 
30 degrees may readily be determined. With this 
transducer orientation, the rays in the main beam are 
only slightly bent by the temperature gradients. Thus 
they strike the bottom at angles that can be cal¬ 
culated directly from the geometry. Furthermore, 
since the rays are only slightly bent, the transmission 
anomaly due to refraction can be ignored. By using 
equation (54) of Chapter 12, then, and by assuming 
A equal to 4 db per kyd, the values of m" at a grazing 
angle of 30 degrees were determined from comparison 
with the observed data. This comparison was made 
on the assumption that the maximum amplitude of 
bottom reverberation on the 30-degree records cor¬ 
responded to scattered sound returning along the 
central raj’ of the main beam; this assumption is 
justified from the qualitative discussion in Section 
15.1. 

The average scattering coefficients determined 
from these analyses of the data in reference 2 are 
shown in Table 2. The 10 log m" values in Table 2 

Table 2. Average values of in" for various bottom areas. 


Bottom type 

10 log m" 
(beam 
horizontal) 

10 log m" 
(beam 
down 

30 degrees) 

k 

COARSE SAND 

-33 

-24 

2.0 

FIXE SAND 

-32 

-26 

1.5 

SAND-MUD 

-25 

-21 

1.0 

MUD 

-29 

-20 

2.0 

ROCK 

-18 

-9 

2.0 

FORAMIXIFERAL SAXD 

-26 




are of course averages of the values obtained from 
the nineteen groups into which the original data were 
subdivided. That is, each group gave a value of 
10 log m" for a definite bottom type, and the entries 
in Table 2 are each averages over all the groups per¬ 
taining to one particular bottom type. 

In interpreting these average scattering coefficients, 
it should be remembered that the 0-degree data were 
obtained with a horizontal transducer near the sur¬ 
face so that an important part of the received bottom 
reverberation reached the transducer along paths re¬ 
flected from the ocean surface. As a result, the values 
of 10 log m " inferred from comparison of the 0-de¬ 
gree data with equation (54) of Chapter 12 are 6 db 
greater than the true value of the bottom scattering 










BOTTOM SCATTERING COEFFICIENTS 


317 


coefficient. The values of 10 log m" shown in the 
first column of Table 2 are true values, that is, they 
are 6 db smaller than the values found from com¬ 
parison of equation (54) of Chapter 12 with the ob¬ 
served reverberation levels. On the other hand, no 
such G-db correction for surface reflection was neces¬ 
sary for 30-degree data, since at the latter transducer 
orientation almost none of the projected sound 
strikes the surface before reaching the bottom. 3 

Having described the procedure for calculating the 
0- and 30-degree columns in Table 2, we now proceed 
to use these entries for an estimate of the angular de¬ 
pendence of m". It will be recalled that for all the 
0-degree entries the grazing angle of the sound on the 
bottom lay between 9 and 13 degrees; for present 
purposes the grazing angle for all the 0-degree entries 
may be taken as 10 degrees. The grazing angle for all 
the 30-degree entries may be taken as 30 degrees, 
since at such a great angle of depression the effect of 
refraction is negligible. Thus for each of the bottom 
types considered, we have in Table 2 an m" for a 
10-degree grazing angle and an m" for a 30-degree 
grazing angle. By assuming a relationship of the 
form 

m" ~ sin 4 (9), 

it is possible to calculate k for each bottom type. The 
resulting values of k, rounded off to the nearest half- 
digit, are displayed in the last column of Table 2. 

These values of k are not too reliable, since in order 
to calculate the individual scattering coefficients in 
Table 2 a number of assumptions were required 
about such questions as the proper method for 
averaging data obtained on different days, and the 
proper comparison between the point and band 
methods of averaging when the reverberation levels 
are changing rapidly. These assumptions, described 
in detail in reference 2, mean that the results of 

a It will be recalled that a 6-db correction was argued in 
Section 12.5 for surface scattering coefficients. It may be 
thought that a similar correction should be applied to bottom 
scattering coefficients, to take account of possible reflections 
in the layer of scattering material at the bottom. This correc¬ 
tion arises, in the case of surface scattering, because the scat¬ 
tered are thought to extend an appreciable distance into the 
water side of the air-water boundary; sound can penetrate the 
scattering layer and strike the air-water boundary at which 
most of the actual reflection takes place. For bottom scatter¬ 
ing, on the other hand, although the bottom scattering layer 
is not infinitely thin, most of it does lie on the solid side of the 
twilight region separating the sea volume from the earth’s 
crust. Thus, there is no need to introduce a correction to 
bottom scattering coefficients due to reflection at the bottom; 
in fact such a correction, if introduced, would have no physical 
significance. 


Table 2 may be somewhat in error. Nevertheless, 
Table 2 does indicate that the value of m" increases 
at least as rapidly as the first power of sin 9 for 
grazing angles between 10 and 30 degrees. 

The data of reference 2 give no information on the 
nature of m"{9) for grazing angles 9 less than 10 de¬ 
grees. It was assumed in reference 1, from which 
Figure 4 was taken, that m" is proportional to sin 2 9 
for angles 9 greater than 9 degrees, and was constant 
independent of 9 for angles 9 less than 9 degrees. The 
solid lines in Figure 4 are the reverberation levels 
predicted on this basis and fit the observed points 
very well, even at the extreme range of 360 yd on the 
lower curve, where the grazing angle is only 4 degrees. 
The very good fit evidenced in Figure 4 seems to indi¬ 
cate that m" is constant independent of grazing angle 
at angles less than 10 degrees. However, there is al¬ 
most no other information on the dependence at 
angles less than 10 degrees; and a constancy of scat¬ 
tering coefficient as the grazing angle decreased be¬ 
low 10 degrees would make the law of scattering a 
very complicated function of angle at these small 
angles. For these reasons it is probably best to regard 
the dependence of m" on grazing angle for angles less 
than 10 degrees as still uncertain. More measure¬ 
ments of this dependence are needed; to obtain values 
of m" at small grazing angles, it will be necessary to 
make measurements in isothermal water. 

Impossibility of Determining m"{9) with 
Horizontal Beams 

We shall now discuss why it was not possible, from 
the 0-degree data alone, to determine the dependence 
of m" on grazing angle on the bottom. Two factors 
are involved: (1) uncertainty in the beam pattern 
correction, and (2) the lack of any large variation in 
the grazing angle, owing to the effect of refraction. 

For horizontal transducers, the beam pattern cor¬ 
rection as determined from equation (41) of Chapter 
12 is small for values of 9 less than 6 degrees (see 
Table 1, Chapter 12). At larger angles the correction 
increases rapidly because of the sharp decline in the 
measured beam pattern at the edge of the main lobe. 
At an angle of 10 degrees, for example, the correction 
is about 20 db. The application of this large correc¬ 
tion appeared to seriously overcorrect the data 
analyzed in reference 2, giving very large values of 
m" at close ranges. It is not difficult to find reasons 
for this inability to calculate m" correctly at angles 
well off the main lobe. In the first place, the very use 
of equation (41), Chapter 12, for the beam pattern 




318 


SHALLOW-WATER REVERBERATION 


correction is questionable at large angles, as has been 
pointed out previously. In addition, the ship pitches 
and rolls; at large angles even a small change in 
orientation of the projector may make a large dif¬ 
ference in the received reverberation. There is the 
further complication that measured beam patterns 
in the vertical plane have not always agreed with 
the measured patterns in calibrating stations. 5 These 
arguments, taken with the overcorrections noted in 
reference 2, suggest that a correction of 10 db or 
more is about the maximum which can be safely 
applied to measured bottom-reverberation levels, if 
accurate values of the bottom-scattering coefficients 
are to be expected. We can conclude that the use of 
equation (41) of Chapter 12 to obtain m" for rays 
leaving the projector at large angles (greater than 
6 degrees) is quite questionable. 

Also, little information about the angular depend¬ 
ence of m" could be obtained from rays leaving the 
projector at angles within the main beam, that is, 
with initial angles less than 6 degrees. For, in these 
experiments the incident angle at the bottom of the 
rays within the 6-degree cone was essentially con¬ 
stant; at best, this grazing angle varied only slowly 
with range. This fact alone would make fruitless any 
attempt to determine, from the data of reference 2, 
a detailed degree-by-degree dependence of m" on 
grazing angle. Furthermore, the value of the scat¬ 
tering coefficient itself, at ranges beyond the region 
where the main beam strikes the bottom, becomes 
more and more doubtful as the range increases, be¬ 
cause of uncertainty in the value of the transmission 
anomaly. These remarks explain why the data of 
reference 2 were capable of giving m" accurately only 
for the angle of incidence on the bottom correspond¬ 
ing to the peak of the reverberation. 

It is worth noting that the virtual constancy of the 
angle of incidence on the bottom, for rays within the 
6-degree cone, should be a rather general result with 
all types of refraction patterns. This conclusion is 
deduced from Snell’s law of refraction, as follows. 

Snell’s law, which was proved in Chapter 2, tells 
us that 

c 

cos d = — cos 9 0 > (4) 

Co 

where 6 is the bottom grazing angle, do is the angle of 
the ray at the projector, c is the velocity of sound at 
the bottom, and c 0 is the velocity of sound at the 
projector. It is clear from equation (4) that the bot¬ 
tom grazing angle will be smallest for the ray which 


leaves the projector at 0 degrees. Thus, the derivative 
dd/ddo equals zero at do = 0; and d necessarily varies 
but little for all the rays leaving the projector within 
a few degrees of the projector axis. 

15.3.2 Dependence on Frequency 

A report by UCDWR 6 presents measurements de¬ 
signed to determine the dependence of the bottom- 
scattering coefficient on frequency. These measure¬ 
ments were made at 10, 20, 40, and 80 kc, with the 
transducers directed downward at an angle of 30 de¬ 
grees with the horizontal. The measurements were 
made in two shallow water areas near San Diego, both 
with rocky bottoms. The ping lengths used were be¬ 
tween 4 and 8 msec. Further details concerning the 
bottom character and the experimental procedures 
are given in reference 6. From comparison of the 
measured reverberation levels with equation (54) of 
Chapter 12, values of 10 log m" were determined at 
each frequency and at each of the two positions where 
measurements were made. These values of 10 log m" 
were obtained assuming the transmission anomaly A 
in equation (54) as zero; because measurements were 
performed in very shallow water, this assumption 
should introduce very little error. The results ob¬ 
tained in reference 6 are tabulated in Table 3. 

An irregular variation of 10 log m" with frequency 
is noted in Table 3, but according to reference 6 this 


Table 3. Backward scattering coefficients (10 log m") as 
a function of frequency at 30-degree grazing angle. 


Frequency in kc 

10 

20 

40 

80 

Area I 

-11 

-6 

-8 

-14 

Area II 

-22 

-17 

-21 

— 15 


variation is less than the estimated error of calibra¬ 
tion. Also, according to reference 6, change in trans¬ 
ducer patterns due to changes in frequency and 
swinging of the ship at anchor could have introduced 
errors compared with which the observed variation is 
not significant. Thus there is no evidence that the 
bottom scattering coefficient for rocky bottoms at a 
grazing angle of 30 degrees has any systematic fre¬ 
quency dependence for the frequency range 10 to 
80 kc. The mean value of 10 log m", averaged for 
10, 20, 40, and 80 kc is —10 + 3 db for position I 
and — 19 ± 3 db for position II. 

The mean values of 10 log m" at a grazing angle 
of 30 degrees, quoted in the preceding paragraph, 
should be directly comparable with the 30-degree 







BOTTOM SCATTERING COEFFICIENTS 


319 


value of m" for ROCK in Table 2. It is seen that there 
is very good agreement with Table 2 for Area I, but 
there is a difference of 10 db between the value of rn" 
(30 degrees) obtained at Area II and the value of m" 
(30 degrees) in Table 2. This difference could be due 
to sampling error; it is estimated later that the 
quartile deviation of m" for areas of similar bottom 
classification is + 5 db. In this regard it is significant 
that reference 6 states that the bottom of Area II 
had patches of SAND-AND-MUD. Thus, it is not 
too surprising that the mean bottom scattering coeffi¬ 
cient in Area II should be lower than in Area I, 
where, according to reference 6, the bottom was 
covered with boulders. 

The results of reference ti give no information on 
the dependence of m" on frequency for other types of 
bott om than ROCK. There is no reason to expect that 
any marked dependence on frequency would be dis¬ 
covered. However, it is necessary to definitely know 
the frequenc 3 r dependence, if any, in order to predict 
the effect on bottom reverberation of varying the 
frequency of echo-ranging gear. Also, knowledge of 
this frequency dependence would enable us to assess 
accurately the present information on the dependence 
of m" on bottom type, much of which is based on the 
assumption that m" does not depend on frequency. 
For these reasons it would be desirable to obtain 
additional measurements over all types of bottom of 
the dependence of m" on frequency. 

15.3.3 Dependence on Bottom 

An analysis of bottom-scattering data obtained 
with horizontal beams is given in reference 3. These 
data include many more records than are analyzed 
in reference 2, among which are data at 10, 20, and 
24 kc. A portion of the data was analyzed in a manner 
similar to that used in reference 2, except that absorp¬ 
tion was not included in the transmission anomaly; 
the transmission anomaly A in equation (54) was 
computed from the refraction pattern alone. This 
analysis gave the results listed in Table 4 for a graz¬ 
ing angle at the bottom of 10 degrees. In Table 4, as 
in Table 2, the values of m" have been corrected to 
the true values. Table 4 should, of course, be com¬ 
parable with the 0-degree column of Table 2, since 
the grazing angle on the bottom (10 degrees) is the 
same for both tables. 

In reference 3, in addition to this analysis, a more 
complicated analysis was also attempted to deter¬ 
mine the variation of the bottom-scattering coeffi¬ 


cient with angle of incidence. Some evidence that in" 
increased with increasing grazing angle was found, 
but it was not possible to decide which of the three 
laws, m" constant, in" proportional to sin 9, or m" 
proportional to sin 2 9, was most nearly representative 
of the bottom scattering. In view of the inconclusive 
nature of the results, and also because this analysis 
rested on some questionable assumptions, these re¬ 
sults for the angular dependence of in" were not in¬ 
cluded in Section 15.3.1. 


Table 4. Average values of 10 log in" for various bottom 
areas for grazing angle at bottom of 10 degrees. 


Bottom type 

10 kc 

20 kc 

24 kc 

COBBLES 

-16 

-16 


ROCK 

-23 

-23 


MUD 

-32 

-38 


MUD 

-31 

-34 


MUD 

-34 

-36 


MUD 



-36 

MUD 



-37 


The value of m" for the ray leaving the projector 
at an angle of 5 degrees was relatively independent of 
the assumptions made. The values of m" for this ray 
are given in Table 5. Table 5 includes, for some of the 
records studied, the grazing angle of the 5-degree ray 
as calculated from the measured BT pattern. It is 
seen that in general the 5-degree ray strikes the bot¬ 
tom at a grazing angle of about 10 degrees; thus 
Table 5 should be comparable with Tables 2 and 4. 

From Tables 2, 4, and 5 we can now determine, for 
each bottom type, the average value of m" for a 
grazing angle on the bottom of 10 degrees. To do 
this, we recall that Tables 4 and 5 were obtained on 
the assumption that the transmission anomaly was 
due to refraction alone, that is, that the absorption 
loss was negligible. However, the values of m" in 
Tables 4 and 5 can be corrected for absorption in the 
following way. It can be assumed, from previous dis¬ 
cussions in this chapter, that on the average the 
ranges at which the data of Tables 4 and 5 were 
evaluated were six times the depth of the projector 
above the bottom. These depths are given for the 
measurements listed in Table 5; for the items in 
Table 4 they can be obtained from Table 2 of refer¬ 
ence 6. Thus, the average absorption loss can be cal¬ 
culated at each frequency for each entry in Tables 4 
and 5, by assuming median values of the attenuation 
coefficient at each frequency (see Figure 17 of Chap¬ 
ter 5). If we increase m" by the average two-way 













320 


SHALLOW-WATER REVERBERATION 


Table 5. Average values of to" for various bottom areas for ray leaving projector at an angle of 5 degrees. 


Bottom type 

Depth of bottom 
below projector 
in yards 

Calculated grazing angle 
for 5-degree ray in 
degrees 

10 log to" for 5-degree ray 





Frequency in kc 




10 

20 

24 

COBBLES 

38 

5.0 

-23 

-19 

. . . 

BOULDERS 

10 

* 

-24 

-21 


ROCK 

20 

* 

-29 

-27 


ROCK 

110 

8.5 

-27 

-28 


MUD 

240 

* 

-36 

-35 


MUD AND SAND 

52 

10.2 

-32 

-35 


FINE SAND 

10 

* 

-33 

-36 


MUD 

275 

10.5 

-31 

-37 


MUD AND SAND 

100 

* 

-38 

-38 


MUD 

195 

10.0 

-35 

-39 


ROCK 

15 

* 



-26 

MUD AND SAND 

53 

* 



-31 

MUD AND SAND 

75 

* 


. . . 

-25 

MUD 

240 

11.8 



-25 

MEDIUM SAND 

215 

* 



-31 

MUD 

230 

11.8 



-25 


* Angle not calculated. 


absorption loss in decibels, the entries in Tables 4 
and 5 will be more or less corrected for absorption 
losses, and the resultant values of m" will be the best 
estimates which can be made from the data of refer- 
ence 6. 

Table 6 shows the mean values of rn" determined 
in this way from the data of Tables 4 and 5. The as¬ 
sumed attenuation coefficients at 10 kc, 20 kc, and 
24 kc in db per kyd were 1.3, 3.2, and 4.0 respectively. 
In Table 6 the mean values for each bottom type were 


Table 6. Mean values of backward scattering coef¬ 
ficient at 10-degree grazing angle. 


Bottom type 

10 log to" 
from 
Table 4 

10 log to" 
from 
Table 5 

10 log to" 
from 
Table 2 

ROCK 

-17 

-24 

-18 

MUD 

-27 

-25 

-29 

SAND-AND-MUD 


-31 

-25 

SAND 


-34 

-30 


determined by averaging the corrected values of m" 
for all three frequencies 10, 20, and 24 kc, giving each 
entry in Tables 4 and 5 equal weight. The justifica¬ 
tion for averaging m" for different frequencies has 
been discussed in Section 15.3.2. 

If data for 10 and 20 kc are not averaged with 24-kc 
values, a large part of the data of reference 3 has to 


be omitted. Another reason for including the 10- and 
20-kc data is the following. Examination of the cor- 
rected values of Tables 4 and 5 shows that the as¬ 
sumptions which were made conceiving the range to 
the reverberation peak and the value of the attenua¬ 
tion coefficient seem to overcorrect m" at 24 kc; that 
is, the corrected values of m" at 24 kc frequently 
tend to be abnormally large, especially in deep water 
where the range to the peak is long. For example, in 
the last two MUD enti'ies in Table 5, the x'ange of the 
peak was estimated to be about 1,400 yd, so that the 
two-way transmission anomaly correction was 11 db. 
This made the coiTected values of 10 log m" for those 
enti’ies equal to —14 db, which is much too high for a 
MUD bottom. The reason that this overcorrection 
occurred is not clear. It might indicate that the at¬ 
tenuation coefficient for the sound returned as rever¬ 
beration is less than the attenuation coefficient de¬ 
termined in transmission runs. However, the data of 
reference 2, which appear quite reasonable, are based 
on an assumed attenuation coefficient of 4 db per 
kyd; in the analysis of volume and surface reverbera¬ 
tion in Chapter 14, median reverberation levels were 
fitted quite well by assuming 4 db per kyd as the value 
of the 24-kc attenuation coefficient. Whatever the rea¬ 
son, the existence of this apparent overcorrection 
makes it desirable to include values of m" for 10 and 
20 kc in the averages based on the data of reference 3, 






























AVERAGE BOTTOM REVERBERATION INTENSITIES 


321 


since at the lower frequencies the corrections for at¬ 
tenuation are not as large. 

In Table 6, cobbles and boulders have been 
grouped under rock; and fine sand, foraminiferal 
sand, and medium sand have all been grouped under 
sand. The last column in Table 6 gives the results of 
averaging a similar grouping of the 0-degree values 
of Table 2, including coarse sand under sand; the 
designations sand-mud and mud-and-sand of refer¬ 
ences 2 and 3 have been replaced by the customary 
SAND-AND-MUD. If all the entries of Table 6 
are averaged with equal weight, we obtain the over¬ 
all averages in Table 7. 


Table 7. Overall mean values of backward scatter¬ 
ing coefficient at 10-degree grazing angle. 


Bottom type 

10 log m" 

ROCK 

— 20+5 

MUD 

-27 + 5 

SAND-AND-MUD 

-28 ± 5 

SAND 

-32 + 5 


The values of Table 7 do not differ significantly 
from other estimates of the mean bottom scattering 
coefficients, also based on the data of references 2 
and 3. In reference 7 it is estimated that the quartile 
deviations of the mean bottom scattering coefficients 
are about 5 db for each bottom type. In view of the 
crudeness of a classification system which includes 
all bottom types in only four categories, a quartile 
deviation of this magnitude is not surprising. Thus 
this estimate of the deviation from the mean has 
been included in Table 7. It must be remembered 
that the values of 10 log m" in Table 7 are true 
values; that is, they were determined by subtracting 
6 db from the values of m" inferred from comparison 
of equation (54) of Chapter 12 with the measured 
reverberation levels. The expected reverberation 
levels with horizontal beams will therefore be 6 db 
greater than the levels that would be predicted by 
the use of equation (54) and the values of 10 log m" 
in Table 7. 

15.4 AVERAGE BOTTOM REVERBERATION 
INTENSITIES WITH HORIZONTAL 
TRANSDUCERS 

Bottom reverberation levels are a function of range 
and water depth and in addition depend on refraction 
conditions, transducer orientation, and bottom type. 


For most practical echo-ranging purposes, however, 
the transducer is oriented so that the transducer axis 
is horizontal, parallel to the ocean surface. Under 
these circumstances, over level bottoms, the data 
which have been presented in this chapter can be used 
to make some prediction of average bottom rever¬ 
beration levels. The results of reference 2, discussed 
in Section 15.3.1, show that under most conditions 
the transmission anomaly due to refraction is negli¬ 
gible at ranges up to and including the range of the 
reverberation peak. The results of references 1, 2, and 
3, described in Section 15.2, all show that in water 
other than isothermal or nearly isothermal the range 
of the reverberation peak tends to be about 6 times 
the depth between the projector and the bottom, and 
that this peak corresponds approximately to the 
range at which the 5- to 6-degree ray from the pro¬ 
jector strikes the bottom. The data of references 2 
and 3, which were discussed in Sections 15.3.1 and 
15.3.3, show that the angle at which this ray strikes 
the bottom usually is about 10 degrees in nonisother- 
mal water. Thus, knowledge of the average value of 
m"at a grazing angle of 10 degrees, coupled with equa¬ 
tion (54) of Chapter 12, enables prediction of the aver¬ 
age height and range of the reverberation peak over 
different bottom types in any water depth. It is of 
course necessary to know the value of A in equation 
(54). However a value of A equal to 4 db per kyd is 
probably a good approximation, and for the short 
ranges at which the reverberation peaks are usually 
observed, deviations in practice from 4 db per kyd 
should not be very significant, except possibly when 
the water is quite deep. 

Once the height and range of the bottom rever¬ 
beration peak are determined, the most significant 
quantity for echo ranging is the rate at which the 
reverberation decays as a function of range past the 
peak. At ranges less than the peak the reverberation 
usually decreases with decreasing range; such ex¬ 
amples as Figure 7 and other similar figures in refer¬ 
ences 2 and 3 show that the bottom reverberation can 
hardly increase with decreasing range as rapidly as 
the expected echo level. 8 At some ranges less than 
the principal reverberation peak (where the main 
beam strikes the bottom) reverberation from side 
lobes can be very high. However, it seems on the 
whole that bottom reverberation is likely to be most 
troublesome at ranges past reverberation peak. 

At ranges past the reverberation peak in noniso- 
thermal water, the results of reference 1 indicate that 
on the average the reverberation falls off at about the 







322 


SHALLOW-WATER REVERBERATION 


inverse fourth power of the range, but that large 
variations from this type of decay are observed. 
Since the expected echo level also falls off at about 
the inverse fourth power of the range, 8 the results of 
reference 1 mean that the possibility of obtaining an 
echo usually depends on the level of the reverbera¬ 
tion peak relative to the expected echo level at the 
range of the reverberation peak. If the reverberation 



100 200 300 500 700 1000 2000 


RANGE TO PEAK IN YARDS 

Figure 8. Expected level of peak of bottom reverbera¬ 
tion as a function of range to peak and bottom type. 

peak is high enough to mask the echo at that range, 
then the echo is not likely to be detected at any range 
past the reverberation peak. Conversely, if the echo 
level is well above the reverberation at the range of 
the reverberation peak, bottom reverberation will 
probably not limit echo ranging at any range. 

The above discussion is not by any means a com¬ 
plete treatment of the problem of echo ranging in 
shallow water. Many factors are involved in deter¬ 
mining the echo-to-reverberation ratio at any range. 
Also, no account has been taken of the fact that at¬ 
tenuation at long range makes the echo level drop off 


much more rapidly than the inverse fourth power of 
the range. However, attenuation also decreases the 
received bottom reverberation levels, so that even at 
long ranges the echo and reverberation levels should 
decrease at roughly the same rate. In general, it can 
be said that the problems involved in determining the 
echo-to-reverberation ratio are so complicated that 
no satisfactory quantitative treatment has ever been 
given, although qualitative discussions have been 
presented in a number of places. 9 It appears then from 
the foregoing discussion that with present informa¬ 
tion the best way to characterize bottom reverbera- 
tionlevels is intermsof the level at the principal rever¬ 
beration peak; this level is determined using equation 
(54) of Chapter 12 and the known value of the bottom 
scattering coefficient for 10 degrees grazing incidence. 

Figure 8 shows the expected average standard re¬ 
verberation level at the reverberation peak, as a 
function of the range to the peak and the bottom 
type, for ordinary 24-kc echo-ranging gear sending 
out a horizontal beam. In preparing this diagram it 
was assumed that the range to the peak is six times 
the depth of the bottom below the projector. The 
absorption was taken to be 4 db per kyd, J b was set 
equal to —19 db for the 5- to 6-degree ray, 10 log m" 
was taken from Table 7, and finally the reverberation 
level for a 100-msec ping was calculated by the use of 
equation (55) of Chapter 12. Although Figure 8 is the 
best average curve which can be drawn with present 
information, the likelihood of deviations in practice 
from Figure 8 cannot be overstressed. In particular, 
Figure 8 is not valid in isothermal or nearly isothermal 
water, when the range to the reverberation peak will 
usually be very different from six times the distance 
between the projector and the bottom. It is also not 
advisable to extend the results in Figure 8 to ranges 
less than 100 vd because at such short ranges it is 
again unlikely that the average assumed relationship 
between range and depth will be valid. It should be 
noted that the curves in Figure 8 incorporate the 6-db 
correction for surface reflections discussed in Section 
15.3.3. Thus, when the echo-ranging transducer is 
deep, 6 db should be subtracted from the values in 
Figure 8 to obtain the expected reverberation levels; 
in such situations surface reflections will not be im¬ 
portant in determining the bottom reverberation 
level. 

In conclusion, we repeat that on the average the 
reverberation in nonisothermal water seems to fall 
off at about the inverse fourth power of the range, at 
ranges past the reverberation peak, but that large 

















AVERAGE BOTTOM REVERBERATION INTENSITIES 


323 


variations from this type of decay may be observed. 
Careful examination of the results of reference 1 
shows that the shape of the decay curve is probably 
not the same over all types of bottoms. In other 
words, the probability of distinguishing an echo at 
long range is different over different types of bottoms, 


even with the same echo-to-reverberation ratio at the 
range of the reverberation peak. A preliminary study 
of the shapes of these decay curves over different 
types of bottoms is being made off San Diego, but 
unfortunately, the results of that study are not avail¬ 
able at this time. 



Chapter 16 


VARIABILITY AND FREQUENCY CHARACTERISTICS 


I n preceding chapters, we have derived theoreti¬ 
cal formulas for the average reverberation inten¬ 
sity, and have compared these formulas with average 
reverberation intensities observed in practice. By 
means of this comparison, we have obtained average 
values of the backward scattering coefficient for 
volume, surface, and bottom reverberation. In this 
chapter w r e shall attempt to analyze the differences 
between reverberations from successive pings sent 
out under apparently the same circumstances. These 
differences may be deviations in amplitude, or devia¬ 
tions in the frequency spectrum of the received re¬ 
verberation. 

There are several reasons for such an analysis. 
First, it is desirable to know just how well the average 
curve may be expected to represent individual rever¬ 
beration curves. Secondly, the deviations from the 
average depend on the type of mechanism giving rise 
to the reverberation; thus, analysis of the deviations 
can give valuable information on the sources of re¬ 
verberation. Finally, such an analysis may easily re¬ 
veal significant differences between the behavior of 
echo fluctuation and reverberation fluctuation since 
the mechanisms producing these two types of fluctua¬ 
tion are undoubtedly somewdiat different. These dif¬ 
ferences in behavior, if w r ell understood, may be 
utilized in methods for improving the recognition 
differential for the echo against a reverberation back¬ 
ground. 

16.1 FLUCTUATION 

In analyzing the amplitude deviations of individual 
reverberation curves from the average, it is con¬ 
venient to distinguish between fluctuation and co¬ 
herence. Fluctuation refers to the deviation from the 
average of the intensity received at a definite time 
following the initial ping. This fluctuation is usually 
measured by the variance 

a p s (/ - j) 2 = p- a? (i) 

where I is the average intensity at a time t seconds 


after midsignal, and 1 2 is the average of the square 
of this intensity. Average values will be designated 
by a bar throughout the remainder of this chapter. 
As discussed in Chapter 12, the average intensity at 
the time t is to be determined by the following proc¬ 
ess. A large number of reverberation records are 
taken under circumstances as nearly identical as 
possible, and the intensity of the reverberation at a 
time t seconds after midsignal is read off each record. 
The average of these intensities is the value referred 
to by the bar. 

Evidently, if all the pings were sent out under 
exactly the same circumstances, the received inten¬ 
sity at time t should be constant, and A/ 2 in equation 
(1) would be zero. However, no two pings occur 
under precisely the same circumstances. There are 
variations in the power output of the projector; varia¬ 
tions in the orientation of the receiver because of ship 
roll; variations in such oceanographic factors as wave 
height, wind force, temperature-depth distribution, 
water depth, and type of bottom material; and over¬ 
all variations in transmission anomaly. Some of these 
sources of reverberation fluctuation can be mini¬ 
mized. The power output of the projector can be 
stabilized to a fraction of a decibel; the ship will not 
roll on a calm day; and the effects of changing wind 
force and bottom character can be eliminated by 
studying only volume reverberation. However, large 
fluctuations remain no matter how much control is 
exercised. These remaining fluctuations in volume 
reverberation are regarded as an inherent property 
of reverberation. 

In the derivation of equation (13) of Chapter 12, 
expressing the time variation of volume reverbera¬ 
tion from a single ping, it was assumed that the re¬ 
verberation is due to the scattering of sound by a 
large number of scatterers in the ocean. Fluctuation 
in the received reverberation is caused by the fact 
that the total reverberation amplitude is the sum of 
the amplitudes received from all the individual 
sources. These individual amplitudes have random 


324 



FLUCTUATION 


325 


phases with respect to each other, and the total 
amplitude is large or small depending on the degree 
of reinforcement or interference between these in¬ 
coming individual amplitudes. 

An expression for the fluctuation of intensity 
caused by the combination of a large number of 
amplitudes of equal magnitude but random phase 
was derived by Rayleigh. 1 The probability that the 
resultant intensity will exceed the value I is given by 

p = (2) 

where I is the average intensity. A derivation of equa¬ 
tion (2) is given in Chapter 7. The actual received 
reverberation is of course a combination of ampli¬ 
tudes of many different magnitudes, because the indi¬ 
vidual scatterers are not all of equal strength, because 
the projector is directional, and because the trans¬ 
mission loss to different portions of the ocean may 
not be the same. However, it can be shown that equa¬ 
tion (2) remains valid, even if all the amplitudes are 
not of equal magnitude, provided only that there are 
a large number of amplitudes of each magnitude, and 
that the number of amplitudes of each magnitude re¬ 
mains essentially constant. Thus, formula (2) is im¬ 
plied by the assumptions used in Chapter 12 to de¬ 
rive the expression for the time variation of the 
volume reverberation from one ping, with the proviso 
that the transmission does not change from ping to 
ping. 

The applicability of the Rayleigh distribution func¬ 
tion (2) has been tested by the University of Cali¬ 
fornia. 2 They first chose sets of ten or more typical 
reverberation records in deep water; all the records 
in a given set were taken under similar conditions. 
The ratio of the observed amplitude to the average 
amplitude was computed for various times on each 
set. All told, 420 values of this ratio were obtained 
for the QB transducer, and 500 values of the ratio for 
the QCH-3. The results are plotted in Figure 1. 
There is apparently no major deviation of either ex¬ 
perimental curve from the theoretical expression (2). 
It appears from Figure 1 that the shape of the 
fluctuation curve does not depend significantly on 
such factors as the directivity pattern of the trans¬ 
ducer, nor on the shape of the pulse sent out; both of 
these factors are different for the two transducers. It 
will be noted that equation (2) predicts this inde¬ 
pendence. Since the times chosen on the records were 
well distributed, and since no effort was made to 
distinguish between surface and volume reverbera¬ 
tion, it appears that the expression (2) applies fairly 


well to all portions of the deep-water reverberation 
versus range curve. 

It is not surprising that formula (2) fits the ob¬ 
served fluctuation of surface reverberation levels 
about as well as it fits the fluctuation of volume rever¬ 
beration. If an assumption that the scattering power 
of the surface remains essentially constant from ping 
to ping is added to the other assumptions used in the 




RATIO OF AMPLITUDE TO AVERAGE AMPLITUDE 


-RAYLEIGH 

-OBSERVED 

Figure 1. Agreement between Rayleigh distribution 
and cumulative distribution of observed reverberation 
amplitudes. 

derivation of equation (38) of Chapter 12, for the de¬ 
cay of surface reverberation, the fluctuation formula 
(2) would be predicted for surface reverberation also. 
Separate tests of the applicability of equation (2) to 
surface and bottom reverberation have been re¬ 
ported, 3 which show that this equation is a reasonably 
good fit to the fluctuation of both surface and bottom 
reverberation levels. These results therefore support 
the assumption that the surface scattering power can 

























326 


VARIABILITY AND FREQUENCY CHARACTERISTICS 


be regarded as a function of certain relatively slowly 
varying physical parameters such as the wind force 
and the wave height. 

The derivation of equation (2) is based on the as¬ 
sumption that the number of independent interfering 
amplitudes is large, or, in other words, that the 
number of scatterers which combine to return sound 
to the receiver at a particular time is large. However, 
since this number of scatterers is proportional to the 
volume of space illuminated by the pulse, it is ap¬ 
parent that the effective number of scatterers de¬ 
pends on the ping length, and cannot be large for 
very short pulses. In fact, with pings sufficiently 
short and directional, the received reverberation at 
any instant will arise from at most one scatterer. 
With such pings the average number of scatterers per 
unit volume could be determined from the character 
of the received reverberation, provided the dimen¬ 
sions of the scatterers are small compared to the 
average distance between them. For, under these 
circumstances, the received reverberation would be 
a series of widely separated echoes from individual 
scatterers; from the spacing of these echoes the 
average number of scatterers per unit volume could 
easily be calculated. 

However, it has not yet been possible, and may 
never be feasible, to reduce the ping dimensions suf¬ 
ficiently to resolve all the individual scatterers re¬ 
sponsible for volume reverberation. With standard 
24-kc gear, some of the larger individual scatterers 
are occasionally distinguishable even when pings as 
long as 5 msec are used. 4 ’ 5 However, as a general rule 
individual scatterers cannot be identified even with 
pings as short as 0.1 msec. It has been pointed out 6 
that the Rayleigh distribution does not apply when 
the effective number of scatterers is small, and that 
it may be possible to determine the average number 
of scatterers per unit volume from the disparity be¬ 
tween the observed distribution function and the 
Rayleigh form (2). The theoretical distribution de¬ 
pends on the assumptions which are made concerning 
the total number of scatterers present. The most 
reasonable assumption is that the number of scat¬ 
terers obeys a Poisson 7 distribution. If so, the proba- 
bilitj' P n (N) that N scatterers are present when the 
average number present is n is given by 

Pn(N) = ~&r m (3) 

V ith this assumption, the expected distribution func¬ 
tion of the reverberation intensity is calculated in 


reference 6, as a function of the average number of 
scatterers in the portion of the ocean illuminated by 
the ping at a definite instant. If this average number 
of scatterers is 10, the deviation of the predicted dis¬ 
tribution function from the Rayleigh distribution is 
of about the same order of magnitude as the devia¬ 
tions of the experimental points from the Rayleigh 
distribution in Figure 1. However, the points plotted 
in Figure l are not sufficiently numerous for a deter¬ 
mination of n. It is estimated in reference 6 that 
4,000 points, all taken with the same ping length, are 
required to say definitely whether an observed distri¬ 
bution more nearly approximates the theoretical 
curve for 10 scatterers or the Rayleigh distribution. 

The average number of scatterers n may also be 
predicted, in principle, from the percentage fluctua¬ 
tion in intensity. If the actual number of scatterers 
obeys the Poisson distribution (3), reference 6 gives 
the following relationship 


A/ 2 

GY 


1 

1 + -• 
n 


(4) 


If the intensity is the resultant of a fixed number n of 
amplitudes whose phases vary at random, it is easy 
to show that 


(/) ! » 


(5) 


It is readily verified by direct integration that for the 
Rayleigh distribution (2), 


A/ 2 

Gy 


i 


( 6 ) 


in agreement with equations (4) and (5) when n is 
infinite. A third alternative is to assume that the 
number of scatterers obeys a Gaussian distribution; 
this would be the case, for example, if the variability 
in the number of scatterers were due primarily to 
accidental variations in the length of the ping emitted 
by the projector. With a Gaussian law for the number 
of scatterers, still another formula would be obtained 

for a7V(7) 2 . 

This discussion of reverberation fluctuation has 
completely neglected the fluctuation due to varia¬ 
bility in such factors as transmission loss, projector 
output, and transducer orientation. With well-func¬ 
tioning equipment such as setup C of Section 13.1.1, 
variations in projector output occur so slowly that 
their effect on the short-term fluctuation of rever¬ 
beration is negligible. Variability in transmission loss 
is known to be large and rapid; therefore, the fluctua¬ 
tion in reverberation levels must be partly a result of 





COHERENCE 


327 


SIGNAL 
LENGTH ‘ 




B 


C 



E^ 



REVERBERATION RECORDS 




«XX X)0<:-cx: 0 ~MX>o»**o«moe: ; :X :: «o; < >:«,: >: 


><XX 'X XX: > 

r 

j: ; 


*o>-o-«cx: :x>-«c 1 _xxX ;XICXI xxc 'x> 


—*—ff —w ^''WXOK k»XH')> 'W < h 



Figure 2. The coherence of reverberation. 


fluctuation of the transmission loss between the pro¬ 
jector and the scattering centers. Transmission fluc¬ 
tuations apparently do not obey a Rayleigh distribu¬ 
tion; the standard deviation of the transmitted ampli¬ 
tude is about 42 per cent of the mean transmitted 
amplitude, in practice, as compared to the 52 per 
cent predicted from the Rayleigh distribution. 8 Ship 
roll is another factor, ignored in this sketchy treat¬ 
ment, which is believed to produce significant fluctua¬ 
tions in reverberation. 9 These neglected factors will 
have to be taken into account before the theory of 
reverberation fluctuation can be considered at all 
adequate. Also, the presence of these other sources of 
fluctuation means that a determination of the average 
number of scatterers giving rise to reverberation, by 
a method such as that proposed in reference 6 would 
not be too reliable even if the distribution function 
for the number of scatterers were known. 

At the present time, average reverberation inten¬ 
sities are determined by averaging between 5 and 12 
pings. The validity of this procedure can be esti¬ 
mated from the magnitude of the variance defined by 
equation (1). It is easy to show that the standard 
deviation of the average intensity of n pings is just 

^ AP/n. If the distribution function is Rayleigh, 
then from equation (6) we have 

AT- = (I) 2 - 


Thus the average intensity I of 5 pings has a standard 
deviation of ±0.45/, or roughly 1.5 db. 

16.2 COHERENCE 

The term coherence applied to reverberation refers 
to a tendency of the received reverberation to occur 
in the form of pulses of the approximate duration of 
the ping length. The possession of coherence means 
that if at any instant the reverberation level is high, 
it is likely that the level will remain high for a little 
while, and that if the reverberation level is low, it is 
likely not to become large in a short time. 

An experimental study of reverberation coherence 
has been made by UCDWR, and reported in Section 
X of reference 2. Ten blobs of about equal amplitude 
were chosen at random from two QB records of equal 
ping length, in the time interval between 1.5 and 2.5 
sec after midsignal. The amplitude was measured at 
intervals of about 0.1 ping length from the middle of 
each blob out to 2 ping lengths on either side. The 
average amplitude at each of the measuring positions 
on the ten blobs was then divided by the average 
amplitude at the middle of the blob and was plotted. 
The resulting graphs, one for each of several ping 
lengths, are given in Figure 2, under the heading 
Coherence Analysis. Time is plotted on the horizontal 
axis; amplitude relative to the amplitude at the 

























328 


VARIABILITY AND FREQUENCY CHARACTERISTICS 


middle of the blob is plotted on the vertical axis. 
The ping length r is given in an upper corner of each 
plot and is marked along the time axis for comparison 
with the blob width of the reverberation. It will be 
observed that the diagram of each blob consists of a 
peak with wings trailing off to either side. It appears 
from the figure that the duration of a reverberation 
blob is very close to the ping length, but that the blob 
is peaked in shape and not rectangular like the en¬ 
velope of the ping. 

The coherence of the reverberation can be de¬ 
scribed mathematically by the correlation between 
the reverberation amplitudes at different times. This 
correlation coefficient p is defined by 

mb) - - /«,)] 

Va/ 2 (<i) Va7~2(4) 

If p is very nearly 1, then a high value of the intensity 
at time h is likely to be associated on the same record 
with a high value of the intensity at time h- If p is 
near zero, a given value of the intensity at time h 
gives no information about the intensity on that 
record at time t 2 . If p is very nearly — 1, a high value 
at h implies a likely low value of the intensity at t 2 . 

It can be shown theoretically that p depends, in 
general, only on the difference in time | h — h |, pro¬ 
vided the average intensities at h and h are not too 
different. 10 If the outgoing ping is square-topped, and 
if the average intensities at the times h and t 2 are 
equal, and if, furthermore, the individual intensities 
follow the Rayleigh distribution (2), then it can be 
shown that the correlation coefficient (7) reduces to 



where a = \ h — h \ and r is the ping length. 11 In 
other words, reverberation levels at times close to the 
center of a blob will be high, but levels at times more 
than a ping length away from the center will bear no 
relation to the intensity at the center of the blob. The 
value of p from the relation (8) is plotted as a func¬ 
tion of a /t in Figure 3. Evidently the result (8) for 
the correlation coefficient explains the dependence of 
the blob width on ping length noted in Figure 2. 

The precise functional dependence of p on a in 
relation (8) depends on the assumption of a rectangu¬ 
lar ping and also depends in part on the assumption of 
a Rayleigh distribution for the individual intensities. 
If, for example, reverberation resulted from echoes 


returned from widely spaced single scatterers, rather 
than from a dense population of scatterers, each re¬ 
verberation blob would reproduce the shape of the 
original ping and p(a) v r ould be unity for a < r. 
However, the result that p is zero when a ^ r should 
be quite independent of the assumed distribution 
function for the reverberation intensities. For, when 
a ^ r the reverberation at time h arises from scatter¬ 
ing in a volume of space which does not overlap the 
volume causing the reverberation at time t 2 . Thus, if 
the reverberation levels from two nonoverlapping 
volumes are independent of each other — and this is 
a reasonable assumption — p will be zero whenever 
a ^ r. In other words, no matter what the distribu¬ 
tion of the reverberation intensities, a decrease of 
blob width with decreasing ping length would be ex¬ 
pected. A corollary to this discussion is that the ob¬ 
served dependence of blob width on ping length does 
not lend any appreciable support to the assumptions 
which led to the Rayleigh distribution. 



PING LENGTH 

Figure 3. Theoretical curve for self-correlation of re¬ 
verberation intensity. 

It is possible to determine all sorts of probability 
coefficients from the observed reverberation records; 
these coefficients can then be compared with compu¬ 
tations based on various assumptions regarding the 
sources of reverberation. However, the labor re¬ 
quired to analyze the observed records is so great 
that this method of investigating reverberation has 
not been considered practical. To illustrate, only one 
very crude attempt has been made to quantitatively 
compare the functional dependence predicted by re¬ 
lation (8) with experiment; the agreement cannot 
be said to have been better than qualitative. Another 
difficulty in this approach is that the theoretical val- 



















FREQUENCY ANALYSIS OF REVERBERATION 


329 


ues of many probability coefficients cannot be com¬ 
puted mathematically. However, a few such coeffi¬ 
cients have been computed on the assumption of a 
Rayleigh distribution for the individual intensities. 
One of these is the joint probability P(h,I 2 ,a) of ob¬ 
taining reverberation intensities h and / 2 at the same 
range on two different pings a time interval a apart. 12 
This coefficient is a function of the average velocity 
of the scatterers relative to the echo-ranging vessel. 
Motion of the scatterers relative to the transducer 
changes the ranges and relative positions of the scat¬ 
terers from one ping to the next, thereby giving rise to 
fluctuation of the measured reverberation levels. 
Similarly, the joint probability of obtaining intensities 
/i and / 2 at two different ranges on the same ping has 
been computed. 13 A general discussion of the signifi¬ 
cance of these various probability coefficients is 
given in reference 10. Other references 14-16 may be 
useful in the analysis of fluctuation and coherence. 

It is worth noting that the measured G- to 7-db 
difference between the average intensity and the 
average peak intensity in a band three ping lengths 
long, referred to in Section 13.2, is another measure 
of the coherence of reverberation. For example, if the 
coherence were very poor the average peak height in 
any finite band would be very large. For in that case 
the reverberation intensity at any instant would be 
very nearly independent of the intensity at any other 
instant; crudely, the band could be divided into a 
large number of intervals in each of which the proba¬ 
bility for a given intensity would follow the simple 
Rayleigh distribution (2) or some similar distribution. 
Since the number of intervals would be large, in¬ 
tensities much higher than average would be ex¬ 
pected to occur at least once in the band three ping 
lengths long. So far, it has not proved possible to 
calculate theoretically, with accuracy, the average 
peak height in a band. 

16.3 FREQUENCY ANALYSIS OF REVER¬ 
BERATION FROM NARROW- 
BAND PINGS 

The received reverberation is often used as a 
reference frequency for the estimation of doppler 
shift in the echo. In order to use reverberation in this 
way, it is necessary to know the average frequency of 
the reverberation and the average frequency band 
width characteristic of reverberation. Such a fre¬ 
quency analysis can also give valuable information 
about the processes giving rise to reverberation. 


The theory of Fourier series tells us that any signal 
of finite duration can be regarded as made up of 
single-frequency components of definite amplitudes 
and phases. A so-called single-frequency ping (CW 
ping) of the sort emitted by ordinary echo-ranging 
gear contains not only the nominal frequency of the 
ping, but also an infinite number of other frequencies 
(see Section 12.5). However, the band width within 
which frequency components of significant amplitude 
lie is usually very narrow for ordinary ping lengths. 11 
The returning reverberation is also composed of a 
band of frequencies. With the assumptions that led to 
the Rayleigh distribution (2), it can be shown that 
the band width of the reverberation equals the band 
width of the outgoing ping. For with these assump¬ 
tions, the shape of the returning signal from any 
scatterer is the same as the shape of the ping. The 
factors which may cause the shape of the returned 
signal from any scatterer to differ from the ping have 
been discussed in Section 12.5; in general these 
factors can be neglected for pings 10 msec or longer. 
Thus the received reverberation, which is simply the 
sum of many such signals, must apparently have the 
same band width as the ping. Other sources of rever¬ 
beration fluctuation, in addition to the fundamental 
randomness of phase leading to the Rayleigh distri¬ 
bution, can increase the band width of reverberation. 
However, it can be shown that these sources can be 
neglected for pings 100 msec or shorter. 

Because of the narrowness of the frequency band, 
the experimental determination of the frequency 
spectrum of reverberation is difficult. Besides, the 
quantity which is measured as a function of frequency 
and called the frequency spectrum is merely the 
energy or intensity contained in a narrow band of 
frequencies; the phases of the individual frequency 
components are never measured. For many pur¬ 
poses, therefore, the measured spectrum is not the 
most useful way of describing reverberation. The 
measured spectrum cannot, for example, give clues 
as to those time variations of reverberation which 
cause it to sound like a signal of wavering pitch. 

The UCDWR has, however, developed a special 
device known as the periodmeter for the analysis of 


a The band width may be defined as the frequency band 
containing half the energy in the ping, or as the frequency 
band within which intensities of spectral components are 
no more than 3 db below the intensity of the midfrequency 
component. For a rectangular ping the band width, defined 
in either manner, is approximately the reciprocal of the ping 
duration in seconds; thus, it would be about 10 cycles for a 
100-msec ping [see equation (63) of Chapter 12]. 




330 


VARIABILITY AND FREQUENCY CHARACTERISTICS 


the rapid frequency shifts characteristic of reverbera¬ 
tion. This device 17 measures the time interval be¬ 
tween successive zeros of the alternating signal fed 
into it. The “instantaneous frequency” of the signal 
is interpreted as inversely proportional to the meas¬ 
ured time interval between successive zeros. This 
instantaneous frequency is recorded against time on 
a cathode-rav oscilloscope (which may be photo¬ 
graphed); the frequency appears as a spot whose 
deflection from a base line is a measure of the 
frequency. 

The periodmeter is designed to function in the 
neighborhood of 800 c; thus reverberation must be 
heterodyned to about this frequency before frequency 



Figure 4. Periodmeter record of 800-cycle oscillator 
tone. 


analysis by the periodmeter is possible. Figure 4 is 
the photograph of a trace obtained by feeding an 
800-c oscillator tone into the periodmeter. Figure 5 
shows the traces of a {ting and its echo from an S-class 
submarine; the way the echo mirrors the frequency 
fluctuations in the ping is interesting. Figures 6A and 
6B are observed traces for volume reverberation and 
bottom reverberation, both showing no indication of 
a definite law for the variation of reverberation fre¬ 
quency with time after midsignal. Such traces are 
typical of most reverberation. In all these traces, in¬ 
crease in frequency is indicated by a smaller (lower) 
amplitude on the trace. 

The periodmeter has been used to analyze rever¬ 
beration signals. The results of this analysis will be 
first summarized and then discussed in more detail. 
Since all these conclusions are based on more than 
1,000 observed points, they have a high degree of 
statistical probability. 

1. There are no obvious systematic changes in 
spectral character during the decay of a single rever¬ 
beration. If the average pitch does change with time 
after midsignal in any systematic way, such a trend 
is masked by the much larger irregular variations in 
pitch. 


2. The averages of reverberation frequency over 
time intervals corresponding to the pitch response 
time of the ear (0.1 sec) show variations large enough 
to render target doppler discrimination of 1 knot or 
less highly unreliable. For target speeds oi 2 knots 
or more, target doppler discrimination appears quite 
reliable. 

3. If the outgoing signals are unintentionally fre¬ 
quency-modulated, because of poor design or malad¬ 
justment of the transmitting equipment, the rms fre¬ 
quency spread in the reverberation is increased. 

4. The frequency spread of the heterodyned rever¬ 
beration depends on the audio output frequency and 
the pulse length. The rms frequency spread A/ was 
well-fitted by the formula 

Af = Kf'i t~ h (9) 

where / is the audio output frequency, t is the pulse 
length, and K is a constant. Pulses of length 92 msec 
and frequency 24 kc produced reverberation, which, 
after being heterodyned to 800 c, had a mean rms 
frequency spread of 21.5 c. 

5. The frequency spread of reverberation does not 
seem to depend on the frequency of the outgoing 
signal. 

6. The mean observed reverberation frequency 
agreed with own-doppler values calculated for the 
axis of the projector. At moderate ship speeds, the 
finite width of the projector beam did not, through 
own-doppler effects, cause marked broadening of the 
beam. 

Most of the above results are quite reasonable. 
The first result indicates that the reverberation arises 
from a process which is essentially random. The 
last result indicates that the mean reverberation 
frequency can be used as a reference for the measure¬ 
ment of target doppler. That is, the mean reverbera¬ 
tion frequency agrees with the following formula 
for the frequency of an echo received on a moving 
ship from a stationary target. 

h = /(l + ~ cos , (10) 

where v is the velocity of the echo-ranging ship, 6 is 
the angle between the projector axis and the line of 
motion of the ship, / is the frequency of the emitted 
ping, and c is the velocity of sound. This agreement 
is theoretically expected. 1S ' 19 

The second result indicates that estimates of target 
doppler are not reliable unless the target is moving at 
speeds of 2 knots or more. The third result, that the 





FREQUENCY ANALYSIS OF REVERBERATION 


331 



Figure 5. Periodmeter record of a ping and its echo. 


observed frequency spread in reverberation should be 
increased by frequency fluctuation in the outgoing 
ping, is also not surprising. 

The dependence of the frequency spread on the 
audio output frequency, described by equation (9), 
seems at first sight rather difficult to understand. In 
principle, heterodyning simply subtracts a constant 
frequency from the frequencies of all components of 
the incoming reverberation signal; thus, if there is no 
distortion, the average frequency spread should not 
depend on the audio output frequency. 2021 However, 
with a little thought, it can be seen that the difficulty 
arises because the time intervals between successive 
zeros of a complex signal are not related in any sim¬ 
ple way to the frequency spectrum of that signal. 
Thus, the assumption that the rms deviation of the 
instantaneous frequency read from the period- 
meter should be exactly equal to the true rms fre¬ 
quency spread of the reverberation spectrum is cer¬ 
tainly not warranted. The complete elucidation of the 
relation between periodmeter readings and the rever¬ 
beration spectrum requires a satisfactory theory of 
the periodmeter, which has not yet been developed 
because of mathematical difficulties. A partial anal¬ 
ysis of the theory of the periodmeter is given in a 
report by UCDWR. 17 There it is predicted that the 
rms spread of the instantaneous frequency should 
obey the formula 

Af = Kf* t~' a (11) 

which differs somewhat from the observational equa¬ 
tion (9). However, the mode of derivation of equa¬ 
tion (11) has been subjected to some criticism. 20 

This difficulty in relating the response of the 
periodmeter to the spectrum of the reverberation 



VOLUME REVERBERATION 



Figure 6. Periodmeter records of volume and bottom 
reverberation. 

illustrates the difficulty which is always experienced 
in predicting the response to reverberation of any 
complex circuit, such as the ear or other doppler 
discriminator. It can be argued that, for the ear, the 
instantaneous frequencies measured by the period¬ 
meter are more significant than the spectrum which 
gives the intensities in very narrow frequency bands. 
However, because of the complexity of the ear, this 
assertion requires proof ; for this reason conclusion 
2 above, and other similar conclusions, cannot be 
relied on if based on evidence from the periodmeter 
alone. 










332 


VARIABILITY AND FREQUENCY CHARACTERISTICS 


16.4 REVERBERATION FROM WIDE¬ 
BAND PINGS 

Up until now, this volume has been concerned 
almost completely with reverberation from narrow- 
band CW pings (in a CW ping the nominal trans¬ 
mitting frequency is fixed during the interval of 
transmission). However, other types of pings have 
been used, as in FM sonar. In this section we shall 
examine the reverberation resulting from the use of 
wide-band pings. 

Strictly speaking, no ping which has a finite dura¬ 
tion can possibly be single-frequency; other fre¬ 
quencies than the nominal one must be present in 
order that the signal can build up and die off. How¬ 
ever, 100-msec CW pings at 24 kchave band widths of 
the order of 10 c, while wide-band pings often have 
band widths of 1,000 c or more (for example, a 
1-msec CW ping has a band width of 1,000 c). It may 
be expected that this very real difference in the order 
of frequency spread will be reflected in a difference 
in the nature of the returning reverberation. 

According to Section 12.5, widening the frequency 
band in the ping should not seriously affect average 
reverberation levels, if the frequency response of the 
gear is sufficiently flat. In other words, the theoretical 
formulas in Chapter 12 giving the average time decay 
of reverberation from a single ping should be just as 
valid (or invalid) for wide-band pings as for narrow- 
band pings. It is true that many of the parameters 
in the formulas of Chapter 12 are frequency-de¬ 
pendent, such as the transmission loss, transducer 
directivity, and the scattering coefficients. However, 
these quantities vary smoothly and relatively slowly 
with frequency and can be replaced by their averages 
over even a 10-kc wide band without introducing 
much error. Thus the resultant theoretical average 
reverberation levels for wide-band pings are simply 
an average, over the frequency band included in the 
ping, of the levels predicted for the narrow-band 
pings. A more quantitative discussion of the qualita¬ 
tive ideas in this paragraph can be found in a report 
by CUDWR. 22 

The average fluctuation, as defined by equation 
(1), cannot be written simply as the average of the 
fluctuations of the individual frequency components. 
Thus, there is reason to expect that the fluctuation 
of wide-band pings may be different from that of CW 
pings. The expected magnitude of the fluctuation of 
wide-band pings will depend on the mechanism 
hypothesized as responsible for the fluctuation. Con¬ 


fining our attention for the moment to those wide¬ 
band pings resulting from the use of very short ping 
lengths, then, if the Rayleigh distribution is valid, it 
is apparent from the form of equation (2) that the 
magnitude of the fluctuation as defined by equation 
(1) does not depend on the ping length. In other 
words, with square-topped CW pings the magnitude 
of the fluctuation will be the same for wide-band 
pings as for narrow-band pings. 

However, the decrease of the ping length required 
to widen the frequency band of a CW ping will in¬ 
crease the rapidity of the fluctuation, since it de¬ 
creases the blob width. This decrease in blob width 
does not necessarily improve the recognizability of a 
returning echo since the echo length is decreased 
correspondingly. Studies of echoes resulting from CW 
pings (see Chapters 21 and 23) indicate that echoes 
look very much like reverberation blobs, of width 
about equal to the ping length. This similarity be¬ 
tween echoes and reverberation blobs makes it very 
difficult to devise means for improving the detectabil¬ 
ity of echoes from CW pings against a reverberation 
background, other than the obvious but not always 
feasible procedure of reducing the average reverbera¬ 
tion intensity. It is true that a time average over a 
relatively short interval will include a large number 
of reverberation blobs and this time average will 
fluctuate much less rapidly than the unaveraged re¬ 
verberation. However, because of the similarity of the 
echo to the reverberation blobs, any averaging pro¬ 
cedure applied to the sound returning from a short 
CW ping is likely to eliminate the echo. 

Nevertheless, by adjusting the time interval over 
which the average is taken, some beneficial effect 
may be hoped for, especially when the echo intensity 
is much larger than the average reverberation in¬ 
tensity. Some studies made with very short (0.3 
msec) unmodulated pings support this hope. It was 
found that the use of these very short pings signifi¬ 
cantly reduced the number of false contacts reported 
and that the maximum range at which a target could 
be identified was not affected. 

When frequency-modulated pings are used, the 
signal sent into the water has a continuously varying 
frequency. With such pings, the received reverbera¬ 
tion at any instant may include a wide band of 
frequencies, depending on the band included in the 
original ping and on the receiving pass band of the 
equipment. Theoretical analysis of the expected fluc¬ 
tuation of the reverberation from such pings is diffi¬ 
cult, but it has been shown 23 that the envelope of the 



REVERBERATION FROM WIDE-BAND PINGS 


333 


reverberation from frequency-modulated pings should 
fluctuate more rapidly than does the echo envelope. 
Observational evidence that the rapidity of rever¬ 
beration fluctuation is increased by the use of fre¬ 
quency-modulated pings is quoted in reference 24, 
and similar observations have been reported in other 
sources. The echo length resulting from a frequency- 
modulated ping is equal to the effective ping length, 
just as with unmodulated pings; the effective ping 
length is itself determined by the pass band of the 
equipment. 

The only accurate quantitative data on the fluctua¬ 
tion of reverberation from frequency-modulated 
pings are those given in reference 25. In that report 
it was found that with frequency-modulated pings 
the mean standard deviation of the reverberation 
amplitude is 33 per cent of the mean amplitude, 
significantly smaller than the 52 per cent value pre¬ 
dicted for a Rayleigh distribution. These measure¬ 
ments were made with pings of length 2, 4, and 8 sec, 
frequency-modulated from 48 to 36 kc. Some results 
on the coherence of FM reverberation are also re¬ 
ported in reference 25. 

The mechanism by which such a decrease in the 
magnitude of the fluctuation could be accomplished 
is difficult to visualize; but the possibility of such 
an effect must be admitted, because our theoretical 
understanding of the problems involved is not 


at all complete. Qualitative reports that frequency 
modulation is effective in reducing the magnitude 
of the fluctuation cannot be trusted, since they 
may be based on the results of time averages 
performed somewhere in the complicated recording 
equipment. 

Recapitulating, in echo ranging the objectionable 
features of reverberation are twofold. Reverberation 
masks the echo; also, reverberation simulates the 
echo, so that false contacts are often obtained. It is 
apparent from this chapter that a great deal of in¬ 
formation still is lacking about such characteristics 
of reverberation as blob shape, cause and rapidity of 
fluctuation, and frequency spread. 

Schemes which have been suggested to suppress 
reverberation may be combined with proposals for 
decreasing the average level of the reverberation by 
decreasing the ping length or by increasing the hydro¬ 
phone directivity. Of course, the possibilities of de¬ 
creasing the ping length or increasing the hydrophone 
directivity are always limited by other practical con¬ 
siderations. Once the ping length has been decreased 
as much as practical, and the transducer directivity 
has been increased to its highest practical value, one 
must resort to devices which do not reduce the 
average energy received in the hydrophone, but in¬ 
stead reduce the instrumental effects of reverberation 
in the detecting mechanism. 



Chapter 17 


SUMMARY 


17.1 DEFINITIONS 

17.1.1 Reverberation 

R everberation is a component of background 
heard in echo-ranging gear, and is distinguished 
from the general noise background by the fact that 
it is directly due to the pulse put into the water by 
the gear. The traveling ping meets not only the 
wanted target, but also myriad small scattering 
centers or other inhomogeneities, each of which re¬ 
turns a tiny echo to the transducer. These tiny un¬ 
wanted echoes combine to make up reverberation. 
Thus reverberation, like the echo, is a sound whose 
pitch is definite and is determined by the frequency 
of the projected pulse. Often echoes which would be 
audible over the remainder of the noise background 
are masked by reverberation. To the operator of echo¬ 
ranging gear, reverberation is evident as a quavering 
ring which sets in as soon as the period of sound 
emission is finished. 

The scatterers producing reverberation may be 
located near the sea surface, in the main ocean 
volume, or in the sea bottom. The reverberations 
produced by these three types of scatterers are called, 
respectively, volume reverberation, surface rever¬ 
beration, and bottom reverberation. This distinction 
is physically meaningful, since these three types of 
reverberation apparently have different properties 
and can be experimentally differentiated from each 
other. 

17.1.2 Reverberation Intensity 

The strength of the sound heard or recorded as 
reverberation depends not only on the intensity of 
the backward scattered sound in the water near the 
receiver, but also on the nature of the receiving gear. 
The intensity of the reverberation actually heard or 
recorded, after the sound in the water has been con¬ 
verted to electrical energy by the receiver, amplified, 


and passed to the ear or recording scheme, is called 
the “reverberation intensity” and is given the symbol 
G. As so defined, G equals the watts output across the 
terminals of the receiving gear. In general, the rever¬ 
beration intensity G is a function of time and is re¬ 
lated to the sound intensity in the water by such 
parameters of the receiver system as receiver directiv¬ 
ity and receiver gain. Since the quantity G depends 
on the gear parameters, the absolute magnitude of G 
is usually not of great significance in studies of the 
intrinsic character of reverberation or of the mech¬ 
anisms producing reverberation. In these studies the 
reverberation level, defined under the next heading, 
is ordinarily employed. 

17.1.3 Reverberation Level 

The reverberation level is the decibel equivalent of 
the reverberation intensity defined in Section 17.1.2, 
expressed relative to a standard which makes rever¬ 
berations measured with different gear exactly com¬ 
parable. Specifically, the reverberation level R' is 
defined as 

R' = 10 log G — 10 log (F-F') (l) 

where F is the projector output at 1 yd in decibels 
above 1 dyne per sq cm, and F' is the receiver 
sensitivity in watts of output for a received rms 
sound pressure of 1 dyne per sq cm. Reverberation 
levels are much more useful than reverberation in¬ 
tensities for comparing measurements made with dif¬ 
ferent systems, since under identical external condi¬ 
tions two different systems sending out pings of the 
same length should in principle give the same rever¬ 
beration level if correction is made for transducer 
directivity. Reverberation levels usually refer to the 
average reverberation intensity G found in a suc¬ 
cession of pings. 

Reverberation intensities are proportional to the 
ping duration. Often it is desirable to convert rever¬ 
beration levels to the levels which would be received 


334 


DEEP-WATER REVERBERATION LEVELS 


335 


using a standard ping length. For this purpose we 
define the standard reverberation level R 

R = R' + 10 log ^) (2) 

where R' is the observed reverberation level with 
ping duration r, and t 0 is a standard ping duration 
usually chosen as 100 msec. 

17.1.4 Backward Scattering 
Coefficients 

By “backward scattering” is meant scattering back 
along the incident ray path. If there is only one ray 
path from the projector to the scatterers, only sound 
which is scattered directly backward can give rise to 
reverberation. Thus the more efficient a portion of 
the ocean is in backward-scattering, the higher will be 
the level of the received reverberation. The efficiency 
of a small volume V of the ocean in scattering sound 
backward is specified in terms of the backward-scat¬ 
tering coefficient m, which is defined by the relation 


where b is the average energy scattered by the volume 
V per second per unit incident intensity per unit 
solid angle in the backward direction. The factor 47 t 
is introduced so that in cases where the scattering is 
the same in all directions, the average energy scat¬ 
tered in all directions per second per unit incident 
intensity will be just mV. 

17.1.5 Fluctuation 

The reverberations from two successive pings 
never reproduce each other exactly. This short-term 
variability is called “fluctuation.” A numerical meas¬ 
ure of fluctuation is provided by the variance of the 
reverberation intensity at a time t seconds after mid¬ 
signal. More specifically, suppose that a large number 
n of successive pings are sent out, that records are 
taken of the n resulting reverberations, and that the 
n intensities at a time t seconds after midsignal are 
read off the records. Then if I is the average of these 
n intensities, and I\, I*, ■••,/„ are the n individual 
intensities, then the fluctuation corresponding to the 
time t seconds after midsignal is measured by the 
variance 

-hii-iy. ( 4 ) 

ni =i 


17.1.6 Coherence 

The term “coherence” applied to reverberation re¬ 
fers to a tendency of the received reverberation to 
occur in the form of pulses or “blobs.” The possession 
of coherence means that if at any instant the rever¬ 
beration level is high, it is likely that the level will 
remain high for a little while, and that if the rever¬ 
beration level is low, it is not likely to become large 
in a short time. The degree of coherence can be de¬ 
scribed mathematically in terms of the correlation 
coefficient p between the reverberation intensities at 
two different times on the same record. 


[/(<l) - /(«!)][/«*) - /(*.)] 

P = / —' — > ~ = • (5) 

v [/(h) - l{k)J V U(k) ~ I{k)J 
The bar signifies an average over many successive 
records. 

17.2 DEEP-WATER REVERBERATION 
LEVELS 

In deep water the reverberation heard at ranges 
past 1,500 yd is almost alw r ays volume reverberation. 
At shorter ranges surface reverberation may exceed 
volume reverberation, if the sea state is sufficiently 
high and the transducer beam is horizontal. Pointing 
a directional transducer downward will usually result 
in the surface reverberation being less than volume 
reverberation at all ranges past 100 yd. 

]7.2.i Volume Reverberation 

The following subsections summarize the known 
information concerning reverberation from the vol¬ 
ume of the ocean. The statements apply only to that 
portion of the received reverberation resulting from 
scattering in the ocean volume; the salient facts 
about reverberation from the sea surface are sum¬ 
marized in section 17.2.2. 

Theoretical Formula for Volume Reverbera¬ 
tion Level 

The expected volume reverberation level R'(t) at a 
time t sec after midsignal is given by the formula 

R’(t) = 10 log -(-10 log m + J v 

- 20 log r — 2A + A i, (6) 

where c 0 is the sound velocity in yards per sec; r is the 
ping duration in sec; m is the volume scattering 









336 


SUMMARY 


coefficient; J v is the volume reverberation index; r is 
the range in yards of the reverberation, r = j^Cot; 
A is the total one-way transmission anomaly to the 
range r; Ay is the one-way transmission anomaly to 
the range r due to the effect of refraction. 

Because of surface reflections, not taken into ac¬ 
count in equation (6), observed volume reverbera¬ 
tion levels with horizontal beams will average about 
3 db higher than the levels predicted by that equa¬ 
tion. 

Dependence on Range 

According to equation (6), if the transmission 
anomaly terms ( — 2A + Ay) can be neglected, and 
if the scattering coefficient to is constant throughout 
the relevant portion of the ocean, then the intensity 
of volume reverberation should decay with the square 
of the range; in other words, its level should drop 
20 db for each tenfold increase in range. In practice, 
this simple inverse-square dependence is only seldom 
observed, because (1) the transmission anomaly 
terms can rarely be neglected at ranges greater than 
1,000 yd; and (2) the value of to often depends on 
position in the ocean. The well-established deep 
scattering layers off San Diego, which apparently 
scatter much more strongly than surrounding regions 
of the ocean, are examples of the dependence of the 
scattering coefficient on position. 

Though detailed agreement with equation (6) is 
almost never observed, volume reverberation does 
tend to decrease rapidly with increasing range, as 
predicted qualitatively by that equation. 

Dependence on Ping Length 

According to equation (6), as the ping length is 
increased the intensity of volume reverberation 
should increase proportionally. Although more data 
are needed, measurements to date indicate that equa¬ 
tion (6) does describe the dependence of reverbera¬ 
tion on ping length. This proportional dependence is 
also predicted theoretically for surface and bottom 
reverberation intensities; for these types of reverbera¬ 
tion also, more data are needed, but apparently the 
theoretical relationship is fulfilled. 

Dependence on Frequency 

The frequency-dependent terms in equation (6) 
are the volume reverberation index J v , the transmis¬ 
sion anomaly terms — 2A + A u and the scattering 
coefficient to. The value of the reverberation index 
can be determined from the pattern function of the 
transducer by means of equation (27) of Chapter 12. 


The transmission anomaly can be estimated by the 
methods described in Chapter 5. Available data in 
the frequency range 10 to 80 kc indicate that on the 
average the scattering coefficient to increases about 
as the first power of the frequency. However, the 
data also do not deny the possibility that m is inde¬ 
pendent of frequency, or that it increases as the 
square of the frequency. 

Magnitude of the Volume-Scattering 
Coefficient at 24 kc 

Observed values of 10 log to, inferred from ob¬ 
served reverberation levels, vary between —50 and 
— 80 db, with —60 db as a typical value. The varia¬ 
tions of 10 log to have not been correlated, in the stud¬ 
ies off San Diego, with variations in any factor other 
than depth in the ocean. The variation of to with 
depth off San Diego has not yet been fully explained, 
but its systematic character seems well established. 
Using a projector pointed straight down, the meas¬ 
ured reverberation off San Diego was found to de¬ 
crease down to a range of 600 or 700 ft, but then is 
frequently observed to rise fairly abruptly, and main¬ 
tain a high value for a depth interval of about 700 ft. 
The inferred values of 10 log to for depths within the 
deep scattering layer are often 15 db or more greater 
than the values of 10 log to at other depths. Some of 
these deep layers of high scattering power persist in 
a given area for periods as long as a month or even 
longer. Although the scatterers in these deep layers 
have not been definitely identified, it seems probable 
that they are of biological origin. 

17.2.2 Surface Reverberation 

The following subsections summarize the known 
information concerning reverberation from the sur¬ 
face of the ocean. This information applies primarily 
to reverberation measured with horizontal beams at 
those ranges where surface reverberation exceeds 
volume reverberation. 

Theoretical Formula for Surface 
Reverberation Level 

The expected surface reverberation level R'(t) at a 
time t seconds after midsignal is given by the formula 

R\t) = 10 log j + 10 log j + J,(0) 

— 30 log r — 2.4, (7) 

where to' is the backward scattering coefficient of the 
surface scattering layer; 9 is the angle at the trans- 



DEEP-WATER REVERBERATION LEVELS 


337 


ducer between the sound returning at time t and the 
horizontal plane; J,(9) is the surface-reverberation 
index corresponding to the angle of elevation 0; and 
the other quantities have the meanings given in the 
section entitled “Theoretical Formula for Volume 
Reverberation Level” with the further specification 
that A is the transmission anomaly along the actual 
ray path to the surface. 

Because of reflections from the air-water interface, 
not taken into account in equation (7), measured 
surface reverberation levels with horizontal beams 
will usually be about 6 db higher than the levels 
predicted by that equation. 

Dependence on Range 

According to equation (7), the surface reverbera¬ 
tion intensity at short ranges, where the transmission 
anomaly 2 A can be neglected, should be proportional 
to the inverse cube of the range, provided m' and 
J s (0) also change negligibly with increasing range. 
This simple inverse cube dependence is observed only 
rarely. When refraction near the surface is sharply 
downward, surface reverberation drops abruptly be¬ 
low volume reverberation at the range where the 
limiting ray dips beneath the surface scattering layer. 
Moreover, the decay of surface reverberation inten¬ 
sity is usually faster than inverse cube even when 
downward refraction is weak or absent. For high wind 
speeds (and therefore high sea states) the decay is 
especially rapid; for wind speeds greater than 20 mph, 
the surface reverberation levels usually drop off 
nearly as the fifth power of the range, and rates of 
decay as high as the seventh power have sometimes 
been observed. Factors which may contribute to this 
unexpectedly high decay rate are: (1) a decrease in 
the surface scattering coefficient m' as the incident 
sound ray becomes more nearly horizontal; (2) at¬ 
tenuation; (3) the sound-shadowing effect of surface 
water waves; and (4) image interference, that is, the 
interference between direct and surface-reflected 
waves. 

Dependence on Wind Force 

The wind-speed dependence of surface reverbera¬ 
tion is most marked at short ranges. At ranges of 
1,500 yd or more, with horizontal beams, the received 
reverberation does not depend on wind speed, and for 
this reason is ascribed to scattering from the volume 
of the sea. At a range of 100 yd, as the wind speed 
increases from 8 to 20 mph, the median reverberation 
level rises steeply in a manner roughly described by 


equation (6) of Chapter 14. With horizontal beams, 
little increase in level has been observed as the wind 
speed increases from zero to 8 mph, or as it increases 
beyond 20 mph. 

Dependence on Frequency 

The frequency-dependent terms in equation (7) 
are the surface-reverberation index ,/.,(0), the trans¬ 
mission anomaly term 2.4, and possibly the surface¬ 
scattering coefficient m'. The value of ./ s (0) can be 
determined from the pattern function of the trans¬ 
ducer, by equations (40), (41), and (42) of Chapter 
12. The transmission anomaly can be estimated by 
the methods described in Chapter 5 of Part I. Un¬ 
fortunately, there are no experimental data on the 
variation of surface scattering coefficients with fre¬ 
quency. 

Magnitude of the Surface-Scattering 
Coefficient 

The magnitude of 10 log m! can be obtained from 
comparison of equation (7) with the measured rever¬ 
beration at any range. Although this process is open 
to criticism, since equation (7) does not describe the 
range dependence of surface reverberation very well, 
it furnishes us with the only information we now 
have on the magnitude of the surface scattering 
coefficient. 

By using equation (7), it appears that the increase 
in surface reverberation at 100 yd as the wind speed 
increases, noted in the preceding subsection, is due to 
increases in the surface scattering coefficient m! as 
the sea becomes rougher. Thus, at a range of 100 yd 
the median values of 10 log m' obtained by comparing 
equation (7) with measured levels are —57 db at 
wind speeds less than or equal to 8 mph, and —22 db 
at wind speeds greater than 20 mph. At 1,000 yd, for 
wind speeds greater than 20 mph, 10 log m' averages 
— 31 db. It does not seem possible according to 
Section 14.2.5 to interpret the reverberation meas¬ 
ured at high wind speeds as a result of scattering from 
a dense layer of bubbles. 

17.2.3 Deep-Water Levels with 
Horizontal 24-kc Beams 

For prediction of deep-water, 24-kc reverberation 
levels with horizontal beams, Figure 31 of Chapter 14 
can be used. This figure shows the highest reported 
reverberation levels, the lowest reported levels, and 
the median levels, for various ranges and wind speeds. 



338 


SUMMARY 


The main import of this figure is that it indicates 
both the expected reverberation level and the possible 
spread in values at any range and wind speed. From 
the upper and lower limits in the figure were inferred 
the values of the surface and volume scattering 
coefficients given in Section 17.2.2. 

17.3 BOTTOM REVERBERATION LEVELS 

The following subsections summarize the known 
information concerning reverberation from the ocean 
bottom. This information is mainly concerned with 
reverberation from horizontally projected beams in 
shallow water. Under these circumstances, after a 
sufficient time has elapsed for the beam to reach the 
bottom, the received reverberation is preponderantly 
bottom reverberation. 

17 . 3 .] Theoretical Formula 

The expected bottom reverberation level R'(t) at a 
time t seconds after midsignal is given by the formula 

R'(t) = 10 log y + 10 log ~~ + J b (0) 

— 30 log r — 2A, (8) 

where m" is the bottom scattering coefficient, 6 is the 
angle at the transducer between the sound returning 
at the time t and the horizontal plane, ./&((?) is the 
bottom reverberation index, corresponding to the 
angle of depression 9, and the other quantities have 
the meanings given in Section 17.2.1. 

With horizontal beams and transducers near the 
surface, observed bottom reverberation levels will 
average about 6 db higher than the levels predicted 
by equation (8), on account of surface reflections. 

17 . 3.2 Dependence on Range 

According to equation (8), the bottom reverbera¬ 
tion intensity at short ranges, where the transmission 
anomaly term 2 A can be neglected, should be propor¬ 
tional to the inverse cube of the range, provided m" 
and Jb{9) also change negligibly with increasing range. 
This simple inverse-cube relationship is almost never 
observed. In the first place, because of the distance 
between the transducer and the bottom, reverbera¬ 
tion from the bottom does not set in until a significant 
time has elapsed after the emission of the ping. 
Usually the reverberation then quickly builds up to a 
peak, corresponding approximately to the time when 
the edge of the main beam strikes the bottom. After 


the peak, the reverberation intensity falls off rapidly, 
usually about as the fourth power of the range; how¬ 
ever, very large deviations from the inverse-fourth 
power decay have been observed. 

The range to the bottom reverberation peak de¬ 
pends on refraction conditions and water depth. In 
isothermal water, the peak is expected at a range 
about 12 times the water depth. When the tempera¬ 
ture decrease from projector to bottom is greater 
than 5 degrees, the peak occurs at a range between 
4 and 8 times the water depth, depending on the 
severity of the downward refraction, with a median 
value of 6 times the depth. 

In general, the quantities m" and J&(0) in equation 
(8) are dependent on range. However, at ranges past 
the reverberation peak, both of these quantities 
usually depend only slightly on range. 

17 . 3.3 Dependence on Frequency 

The frequency-dependent terms in equation (8) 
are the surface reverberation index Jb{9), the trans¬ 
mission anomaly 2 A, and the bottom scattering 
coefficient m" . The value of Jb(9) can be determined 
from the pattern function of the transducer, by equa¬ 
tions (53), (41), and (42) of Chapter 12, and, as be¬ 
fore, the transmission anomaly can be estimated by 
the methods of Chapter 5. Measurements on rock 
bottoms indicate no dependence of the bottom-scat¬ 
tering coefficient m" on frequency, in the frequency 
range 10 to 80 kc. It is probable that other bottoms 
as well would show no dependence of m" on the 
frequency of the incident sound, although more data 
are needed to confirm this point. 

17 . 3.4 Dependence on Bottom 

Bottom reverberation levels are not the same over 
all types of bottoms. Although wide variations are 
observed, in general the highest reverberation levels 
are observed over ROCK, lower values over MUD 
and SAND-AND-MUD, and the smallest values 
over SAND bottoms. These classifications of bottom 
type depend on the particle size in the material com¬ 
posing the bottom and are more fully described in 
Chapter 6. 

17 . 3.5 Bottom Scattering Coefficients 

at 24 kc 

The backward scattering coefficient depends on the 
angle at which the sound is incident on the bottom. 



FUTURE RESEARCH 


339 


For a grazing angle of about 10 degrees, a typical 
value for the angle at which sound in the main beam 
strikes the bottom, average values for 10 log m" are 
-20 db for ROCK, -27 db for MUD, -28 db for 
SAND-AND-MUD, and -32 db for SAND. Over 
individual bottoms of a given type, deviations of 
+ 5 db from these average values may be expected. 

There is not much information concerning the de¬ 
pendence of m" on grazing angle. It appears that for 
angles between 10 and 30 degrees m" is roughly 
proportional to the square of the grazing angle. 

17.3.6 Bottom Reverberation Levels 
with Horizontal 24-ke Beams 

Figure 8 of Chapter 15 shows the expected rever¬ 
beration level at the bottom reverberation peak, as a 
function of bottom type and of bottom depth below 
the projector. The height of the peak is a significant 
quantity in assessing the importance of bottom rever¬ 
beration in any given situation. For detailed predic¬ 
tion of the levels at ranges past the peak, accurate 
knowledge is needed of the transmission of sound 
along the various ray paths to the bottom. 

17.4 FLUCTUATION AND FREQUENCY 
CHARACTERISTICS 

17.4.1 Fluctuation 

The measured reverberation is probably the result¬ 
ant of a combination of a large number of small 
amplitudes of random phase. If so, the probability P 
that the reverberation intensity will exceed the value 
I is given by the formula 

p = e -(//D (9) 

where I is the average intensity. For the distribution 
defined by equation (9), the variance defined by equa¬ 
tion (4) is I. 2 Measurements indicate that equation 
(9) is a fairly good description of the distribution of 
reverberation intensities. However, the observed fluc¬ 
tuation of reverberation intensity must, in some part, 
be due to variability in such factors as transmission 
loss and transducer orientatiou. 

17.4.2 Coherence 

Analysis of reverberation records shows that the 
reverberation tends to occur in the form of pulses or 
“blobs” of about the length of the ping. For square- 


topped pings and the intensity distribution defined 
by equation (9), the correlation coefficient in equa¬ 
tion (5) has the value given by 

i(i - -Y for a < r 
P = v rJ - (10) 

(o for a ^ t 

where r is the ping length, and a = — t 2 \. 

17.4.3 Frequency Spread 

For many purposes it is desirable to know the 
frequency spectrum of reverberation, which gives, as 
a function of frequency, the energy in each 1-c band. 
If the reverberation is simply the combination of a 
large number of individual echoes, each with the same 
frequency spectrum as the emitted ping, then the 
resultant reverberation should also have the same 
spectrum as the ping. This conclusion is probably 
not far wrong, although precise measurements of the 
frequency spectrum of reverberation have not often 
been attempted. 

The distribution of the instantaneous frequencies 
of the reverberation (defined in Section 16.3) is also 
useful information. This distribution can be measured 
by an instrument known as the “periodmeter.” 
Periodmeter measurements indicate, among other 
things, that the spread of instantaneous frequencies 
in the heterodyned reverberation depends on the 
audio output frequency and the pulse length, but 
does not depend on the frequency of the outgoing 

ping- 

17. 4.4 Wide-Band Pings 

The fluctuation of the reverberation with wide¬ 
band pings is probably not very much different in 
magnitude from the fluctuation with narrow-band 
pings. However, the rapidity of the fluctuation is in¬ 
creased as the frequency band is widened. In general, 
the average reverberation levels are not affected by 
widening the frequency band of the outgoing ping. 

17.5 FUTURE RESEARCH 

Reverberation studies are a powerful tool in the 
investigation of properties of the ocean. Information 
from such studies is necessary to determine definitely 
the nature of the scatterers, is useful in evaluating 
theories of transmission loss, and can cast light on the 
temperature microstructure of the ocean. Also, these 



340 


SUMMARY 


studies are needed to fill in the gaps in our knowledge 
of the reverberation levels to be expected under vari¬ 
ous conditions. In general, measurements of volume 
reverberation will cast the most light on the funda¬ 
mental properties of the ocean. Experiments of the 
following sort are indicated: 

1. Measurements of reverberation over a very 
wide range of frequencies, from sonic frequencies up 
to several hundred kilocycles. 

2. Measurements of the dependence of volume re¬ 
verberation on transducer directivity, which would 
help evaluate the importance of multiple scattering. 

3. Careful correlation of measured volume rever¬ 
beration levels with simultaneous measurements of 
temperature microstructure. 

4. Careful correlation of measured reverberation 
levels with observed transmitted sound levels, espe¬ 
cially with such features as sound penetration into 
predicted shadow zones. 

5. Reverberation measurements with deep pro¬ 
jectors to demonstrate any fundamental differences 
between the upper and lower layers of the ocean. 

6. A thorough investigation of the deep scattering 
layers, including the use of underwater photography. 

7. Measurements of reverberation in large fresh¬ 
water lakes. 

8. More complete studies of the dependence of re¬ 


verberation on ping length, especially with very short 
pings. 

9. Investigation of various probability and correla¬ 
tion coefficients of the sort discussed in Chapter 16. 

10. Measurements of the dependence of surface 
and bottom scattering coefficients on the grazing 
angle of the incident sound. 

11. Correlation of measured surface reverberation 
levels with simultaneous measurements of optical 
transparency and of entrapped air or other ma¬ 
terial. 

Theoretical investigations of various questions are 
also required, so that the results of these experiments 
may be correctly interpreted. Most of these theo¬ 
retical investigations will be of importance in the 
subject of transmission as well as reverberation. 
Typical subjects for theoretical research would be the 
reflection of sound from a rough surface, and scat¬ 
tering of sound by thermal microstructure. 

A final subject of great importance, which requires 
both theoretical and experimental research, is the 
development of instrumental means for recording and 
computing various time averages which are of interest 
in reverberation studies. Such instrumental pro¬ 
cedures would greatly reduce both the time and ex¬ 
pense involved in the suggested experiments listed 
above. 



PART III 


REFLECTION OF SOUND FROM SUBMARINES AND 


SURFACE VESSELS 



























Chapter 18 

INTRODUCTION 


O bjects may be detected by the echoes they re¬ 
turn. In water, sound waves are absorbed and 
scattered very much less than radio or light waves. 
Consequently, sound waves are particularly useful in 
detecting distant objects under water by means of 
echo-ranging, that is, sending out a sound signal and 
listening for a returning echo. 

The loudness of an echo depends on how much 
sound is absorbed and how much sound is reflected. 
As a signal is sent out, the energy spreads; some of 
it is immediately absorbed by the water and is dissi¬ 
pated as heat energy. The transmission and absorp¬ 
tion of underwater sound have been studied exten¬ 
sively in subsurface warfare, and are described in 
Chapters 1 to 10 of this volume. Some of the energy 
is scattered at random back to the echo-ranging pro¬ 
jector, either by particles or other inhomogeneities in 
the water, or by the ocean surface or bottom. This 
scattering gives rise to a phenomenon known as re¬ 
verberation, which has also been investigated in de¬ 
tail and is treated in Chapters 11 to 17 of this volume. 
The sound distinctly reflected from an obstacle or 
target in the path of the sound beam — such as a sub¬ 
marine or whale — gives rise to an echo. Chapters 
18 to 25 discuss the reflection of sound from vari¬ 
ous underwater targets. 

Many types of targets are encountered in practice. 
In particular, recognizable echoes have been received 
from schools of fish, whales, patches of kelp and sea¬ 
weed, and from sunken wrecks or prominent irregu¬ 
larities on the ocean bottom in shallow water. Certain 
water conditions give rise to echoes; at very short 
ranges, echoes from ocean swells have been observed. 
Wakes, “pillenwerfer,” and other types of bubble 
screens afe effective targets. Their acoustic properties 
have been studied both theoretically and experi¬ 
mentally, and are described in Chapters 26 to 35. In 
addition, icebergs have been detected by echo-rang¬ 
ing, although no such echoes have been measured. 

18.1 TARGET STRENGTH 

In subsurface warfare, submarines, surface vessels, 
and underwater mines are the most important tar¬ 


gets. The reflection of sound from submarines and 
surface vessels has been investigated in terms of 
target strengths, a quantitative measure of their re¬ 
flecting characteristics. Submarine target strengths 
have been studied as a function of the §ize and shape 
of the submarine, its orientation with respect to the 
echo-ranging projector, the distance from the sub¬ 
marine to the projector, and the frequency of the 
echo-ranging sound. Chapters 18 to 25 summarize all 
available information along these lines. 

Echo-ranging measurements on submerged sub¬ 
marines under more or less controlled conditions have 
resulted in a large collection of target strength data. 
In addition, submarine target strengths have been 
computed theoretically and measured in experiments 
with scale models. Unfortunately, very little is known 
about the reflection of sound from surface vessels, as 
no exhaustive series of tests has been made with this 
sole object in mind. The data describing surface 
vessel target strengths are few and scattered; con¬ 
clusions are tentative and uncertain. No measure¬ 
ments have been made of the reflecting characteristics 
of mines. However, spheres of various sizes have been 
frequent experimental targets, and the results of 
echo-ranging measurements on spheres are probably 
applicable to small-object location and the detection 
of mines. 

18.2 USES 

Tactically, knowledge of how submarines and sur¬ 
face vessels reflect sound is very important. A quan¬ 
titative evaluation of the contribution which the re¬ 
flecting characteristics of the target make to the re¬ 
ceived echo intensity is necessary in order to predict 
maximum echo ranges accurately. In submarine 
operations, the reflecting properties of the submarine 
should be known so that effective evasive maneuvers 
may be taken to reduce, as far as possible, the chance 
of sonar contact by enemy antisubmarine vessels. 
For example, it is known that the strongest echo is 
obtained when the submarine presents its beam to 
the echo-ranging vessel. Therefore, keeping the at- 


343 


344 


INTRODUCTION 


tacking vessel off the beam of the submarine is im¬ 
portant in reducing the maximum range at which the 
enemy vessel can detect an echo from it. It should 
also be useful for submariners to know under what 
conditions a submarine is most vulnerable to contact 
by echo ranging, that is, in what position, at what 
aspect, or depth, or speed. In addition, any counter¬ 
measures designed to reduce the probability of con¬ 
tact, such as by making the submarine a less effective 
acoustic reflector, require a quantitative knowledge 
of the reflecting characteristics of the submarine. 

Similarly, such knowledge would be useful to anti¬ 
submarine vessels in suggesting searching or attack¬ 
ing operations. It is also required for the efficient 
design and operation of many underwater echo-rang¬ 
ing devices, such as certain decoys or mines. Informa¬ 
tion on how much sound mines will reflect is impor¬ 
tant both in the design of mine detection gear and 
in the evaluation of echo-ranging equipment tests 
carried out with a particular type of mine. 


This report emphasizes how submarines and sur¬ 
face vessels reflect sound under field conditions. 
Chapter 19 introduces the concept of target strength, 
defines it in terms of quantities directly measurable, 
and derives an expression for the target strength of 
a perfectly reflecting sphere on the basis of ray 
acoustics. Chapter 20 presents the theoretical back¬ 
ground of reflection and scattering of sound from 
bodies of various shapes on the basis of wave 
acoustics, and reviews the theoretical calculations of 
the target strength of a submarine. The technique of 
the direct, field measurements of submarine target 
strengths are described in Chapter 21, the indirect 
measurements in Chapter 22, and all the results of 
submarine target strength measurements are sum¬ 
marized and discussed in Chapter 23. Finally, Chap¬ 
ter 24 comprises all available information on surface 
vessel target strengths and Chapter 25 summarizes 
briefly the reflection of sound from both submarines 
and surface vessels. 



Chapter 19 


PRINCIPLES 


W hen a target is in the path of a sound beam, 
the intensity of the reflected sound measured 
some distance away will, in general, depend on many 
factors, such as the intensity of the sound striking 
the target, the distance from the target to the point 
where the echo is measured, and the size, shape, and 
orientation of the target. Often it is desirable to 
separate these different factors so that the effects of 
the size, shape, and orientation of the target may be 
discussed independently of all other factors. 

Such a separation is possible only when the radii 
of curvature of the sound waves striking the target 
and returned to the receiver are both much larger 
than the dimensions of the target, in other words, 
when the waves incident on the target and the waves 
reflected back to the receiver are essentially plane. In 
terms of ray acoustics, the incident sound rays must 
be substantially parallel over the area of the target 
which they strike, and the reflected sound rays 
must be parallel over the area of the face of the re¬ 
ceiver. 

In this chapter target strength is defined quanti¬ 
tatively in terms of the echo level, the source level, 
and the transmission loss. Then the target strength 
of a sphere is derived as a function of its radius. 
Finally, the effect of pulse length on target strength 
is examined for a simple case. Ray acoustics is em¬ 
ployed throughout the chapter, and the arguments 
are necessarily idealized. No account is taken of the 
wave character of sound; in other words, all effects 
attributable to the wave nature of sound such as in¬ 
terference, diffraction, and phase differences are ex¬ 
plicitly ignored. The conditions under which this ap¬ 
proximation is valid are discussed in Section 19.4. A 
more detailed theory of target strength in terms of 
wave acoustics is presented in Chapter 20. 

19.1 DEFINITION OF TARGET STRENGTH 

Let I 0 be the intensity of the incident sound strik¬ 
ing a stationary target, and I r the intensity of the re¬ 
flected sound measured at some particular point. If 


/o is doubled, I T will also be doubled, other factors 
remaining unchanged; that is, the intensity of the 
reflected sound will be directly proportional to the 
intensity of the incident sound. 

For a given value of I 0 , I T will depend on the 
orientation of the target relative to the incident 
sound and also on where the echo is measured. This 
dependence of I r may be quite complicated. In practi¬ 
cal echo ranging, however, the problem is simplified 
because the echo is always measured back at the 
source — in other words, it is always measured in the 
same direction as the projected sound, and the target 
strength depends only on the orientation of the target. 
Therefore, it will be assumed throughout this chapter 
that the echo is measured at the source. Although this 
admittedly is not the most general case, it is the only 
case of practical importance for echo ranging. 

19 . 1.1 Inverse Square Transmission 
Loss 

The dependence on distance, although complicated 
near the target, becomes very simple far away from 
the target, if the sound rays are assumed to travel in 
straight paths in an ideal medium, with boundaries so 
far away that their effects on sound propagation can 
be neglected. It has been shown in Section 2.4.2 that 
the intensity of sound from a point source, in this 
ideal case, falls off inversely as the square of the 
distance from the source. This same inverse square 
law applies to sound reflected from any target at 
distances much larger than the dimensions of the 
target, since at such distances the target behaves as 
a point source of sound. 

Why the inverse square law holds for the intensity 
of sound reflected from any target, at large but not at 
small distances, may best be understood by studying 
Figure 1. Here are shown rays reflected from a target, 
A to a point near the target, and B to a point far 
away from the target. Rays reaching a point near the 
target come from different points on the target and 
from various directions, if the surface is irregular. 


345 


346 


PRINCIPLES 




Therefore, the way in which the sound intensity near 
the target varies from point to point is complicated. 
Rays reaching a point far away from the target all 
come from essentially the same direction, no matter 
from what part of the target they are reflected. Thus 
the target “looks like” a point source, and the in¬ 
verse square law of intensity will hold. This conclu¬ 
sion, based solely on ray acoustics, is reinforced by 
considerations of wave acoustics, mentioned in Sec¬ 
tion 19.4 and described in more detail in Chapter 20. 

Sufficiently far away from the target, then, I T will 
be not only directly proportional to 7 0 but also in¬ 
versely proportional to the square of the distance r, or 

Ir = A- (1) 

r- 

Here k is a constant which in general depends on the 
size, shape, and orientation of the target. It does not 
depend on the strength of the sound striking the 
target, or on the distance from the target, provided 


I r is measured far enough away from the target to 
make certain that the intensity of the reflected sound 
will follow the inverse square law. Incidentally, this 
relation is not valid for explosive sound, which is 
treated in Chapters 8 and 9. 

Now according to equation (89) in Chapter 2, the 
intensity 7 0 of the incident sound striking the target 
is equal to the intensity F of the projected sound I yd 
away from the source, divided by the square of the 
distance r from the source to the target, provided 
that r is much larger than the dimensions of the 
source. Then 

F 

h = -• ( 2 ) 

r- 

Substitute equation (2) into equation (1), and 



Equation (3) is particularly interesting because it 
shows that, for an ideal medium, the intensity of an 
echo is inversely proportional to the fourth power of 
the range, as long as the echo is measured at the 
source and the range is much larger than the dimen¬ 
sions of the target or source. If logarithms are taken 
and equation (3) expressed in decibels, 

10 log I r = 10 log k -f- 10 log F — 40 log r. (4) 

19.1.2 General Transmission Loss 

All these equations are derived on the assumption 
that the medium through which the sound travels is 
ideal, that all the sound is transmitted freely without 
refraction, absorption, or scattering, and that the 
boundaries of the medium are so far away that their 
effects on the propagation of sound waves may be 
neglected. In other words, as the sound travels each 
way, its intensity falls off according to the inverse 
square law alone. The drop in intensity each way, in 
decibels, is the transmission loss 77, which for this 
ideal case is simply 20 log r. The total transmission 
loss 277 to the target and back again is then 40 log r. 

Generally, however, the intensity of transmitted 
sound under water does not fall off according to the 
inverse square law alone. Sound is absorbed and 
scattered in sea water. It may be bent by tempera¬ 
ture gradients and consequently focused or spread 
out. Often the surface and bottom of the ocean sig¬ 
nificantly affect both transmitted and reflected sound. 
Therefore, 77 will seldom exactly equal 20 log r, and 








DEFINITION OF TARGET STRENGTH 


347 



I 2 3 5 7 10 2 0 30 50 70 100 200 300 500 700 1000 

RADIOS IN YAROS 

Figure 2. Target strength of a sphere. 


the two-way transmission loss is more conveniently 
represented by the more general function 2 H than 
by 40 log r. Equation (4) then becomes 

10 log I r = 10 log k -f- 10 log F — 2 H. (5) 

The total transmission loss 2 II cannot be predicted 
or estimated very reliably because of widely varying 
oceanographic conditions. Instead, it must actually 
be measured during the course of the experiment. 

19 .1.3 Fundamental Definition 

By defining the target strength T as 10 log k, the 
echo level E as 10 log I r , and the source level S as 
10 log F, equation (5) becomes 

T = E - S + ‘111 (0) 

where 2 H is the total transmission loss in decibels 
from the source out to the target and back to the 
source again. Equation (6) is the fundamental defini¬ 
tion of target strength. This equation is always used 
in the computation of target strengths measured at 
sea since it involves only directly measurable quanti¬ 


ties — that is, echo level, source level, and transmis¬ 
sion loss from the source out to the target and back 
to the source. 

Since 7 0 and / r are measured in the same units, it is 
evident from equation (1) that k has the dimension 
of an area and the value of T will depend on the units 
which are used. Since the yard is used in range- 
prediction work as the unit of length, the source level 
is defined in terms of the intensity at 1 yd, and the 
transmission loss, which enters twice into equation 
(6), is defined in terms of the intensity drop from a 
range of 1 yd out to the range of the target. Conse¬ 
quently k in equation (3) may be expressed in square 
yards. 

Equation (6) was derived from physical concepts 
in order to express as a sum of separate terms the 
effects on the strength of the received echo of (1) the 
size, shape, and orientation of the target; (2) the 
intensity of the source; and (3) the range of the 
target. This separation can be realized only at long 
ranges, where the sound reflected from the target be¬ 
haves as if it were emitted from a point source and 
the target strength becomes independent of the 



























































































348 


PRINCIPLES 



range. At long ranges, then, only the transmission 
loss term depends on the range. 

At short ranges, however, the target strength de¬ 
pends on the range as well as on the size, shape, and 
orientation of the target (see Section 20.4.4). If the 
source is so close to the target that different parts of 
the target are struck by sound of different intensities, 
or if the receiver is so close that the spreading of the 
sound reflected from the target to it is not the same 
as the spreading from a point source, the target 
strength term will depend on range. Therefore, at 
short ranges equation (6) does not serve primarily to 
separate the effects of range, transmission conditions, 
source level, and target characteristics on the echo 
level, but rather to define target strength under the 
particular conditions of that measurement. 

19.2 TARGET STRENGTH OF A SPHERE 

Because a sphere is perfectly symmetrical, the 
echoes which it returns to a sound source are com¬ 


pletely independent of its own orientation. For this 
reason, spheres are convenient targets and have fre¬ 
quently served as experimental targets in echo-rang¬ 
ing measurements. In this section, the target strength 
of a sphere will first be derived simply and intuitively 
by considering the total intercepted and reflected 
energy without regard to the angular distribution of 
energy within the reflected sound beam. Then a more 
rigorous derivation — within the framework of ray 
acoustics — will be presented, in which the angular 
distribution of the reflected energy is considered in 
detail. 

19.2.1 Simple Derivation 

Consider a plane wave of sound of intensity I 0 
striking a sphere of radius A and cross-sectional area 
xA 2 . Then the total sound energy intercepted by the 
sphere per unit time will be ttA 2 I 0 and, if reflection is 
perfect, the total sound reflected from the sphere per 
unit time will also be irA' 2 I 0 . 











TARGET STRENGTH OF A SPHERE 


349 


Now assume that this sound energy is reflected uni¬ 
formly in all directions. At a distance r from the 
center of the sphere, it will be spread uniformly over 
the surface of a sphere of radius r or over the surface 
area 4xr 2 . Since the intensity I T of the reflected sound 
equals the total energy xA 2 / 0 reflected by the target 
sphere per unit time, divided by the area 4xr 2 over 
which it is distributed, then at a distance r from the 
sphere 


Ir 


4 xr 2 



(7) 


But, from equation (1) 



( 8 ) 


where r is the distance from the target to the point 
where the echo is measured. Therefore by substitu¬ 
tion 

, A 2 

k = j (9) 

and T = 10 log k = 20 log (10) 

where T is the target strength and A the radius of the 
sphere. With the yard chosen as the unit of length, 
A becomes the radius of the sphere in yards, and from 
equation (10) it is evident that the target strength is 
the echo level of the target in decibels above the echo 
level from a sphere 2 yd in radius. Target strengths 
for spheres of various radii are shown in Figure 2. 


19.2.2 Rigorous Derivation 

This derivation explicitly assumed that sound is 
reflected from a sphere uniformly in all directions. To 
justify this assumption, consider the same sphere of 
radius A in Figure 3. Two adjacent rays x and y 
separated by a distance dz are traveling parallel to 
00' and strike the sphere at X and Y respectively, 
making angles of 0 and 0 + dd with the sphere radii 
drawn to the points X and Y. From Figure 4, 

dz = XY cos 0 = A cos Odd. (11) 

Now rotate rays x and y about 00'. These rays 
will describe circular cylinders and dz will generate 
an area ds between them, where 

ds = 2 nXWdz = 2 x ( A sin 0) (A cos Odd) (12) 
or ds = 2 ir A 2 sin 0 cos Odd. (13) 

The total energy dJ striking the sphere at angles 
between 6 and 0 + dO will be the product of the in- 




K-* —1 


B 



h—z —1 


ECHO 



w=i+2i = 3 x = 3z 



—1 * 1— C 



h— 1 H 


ECHO 


r = x + 2 2=2 z 


Figure 4. Effect of pulse length on target strength, 
echo length, and echo structure. 


tensity 7o of the incident sound and the cross-sec¬ 
tional area ds, or 

d.J = I 0 ds = 2irA 2 I 0 sin 0 cos Odd. (14) 

Now consider the reflected rays x' and y' making 
angles of 20 and 20 + 2 dO with 00' in Figure 3. At a 
distance r from the center of the sphere, x' and y' will 
be separated by a distance dZ. At a distance much 
larger than the radius of the sphere, dZ becomes much 
greater than XY ; and x' and y' may be replaced by r. 

dZ = r 2 (dO) = 2rdO. (15) 

Again rotate the rays about 00', and dZ will generate 
the area dS between them, where 

dS = 2-irPQdZ = 2 x(r sin 20) (2 rdO) (16) 
















350 


PRINCIPLES 


or dS = 4irr 2 sin 2 Odd = 8irr 2 sin 0 cos Odd. (17) 

Since the intensity of the reflected sound equals the 
energy reflected per unit time divided by the area 
over which it is distributed, then 

dJ 2 ttA 2 sin 0 cos ddd A 2 

Ir = — = -—-— 1 0 = —JO- (18) 

dS 8 rr 2 sm 0 cos ddd 4r 2 

Thus I T is independent of 0 and therefore is inde¬ 
pendent of the direction of the reflected sound, and 
equation (18) is identical with equation (7) derived 
from a simpler analysis. Rigorously, then 

T = 10 log k = 10 log ^ = 20 log , (19) 

where T is the target strength and A the radius of 
the sphere. 

Equation (19) applies only to target strengths 
measured far away from the sphere. Close to the 
sphere, the target strength will also depend on both 
the direction 0 and the range r. 

19.3 EFFECT OF PULSE LENGTH 

So far it has been tacitly assumed that continuous 
sound strikes the target and is reflected back to the 
projector. Usually, however, sound pulses of finite 
length are sent out, and most target strengths are 
measured with such sound pulses. In general, target 
strength will be a function of pulse length, and the 
dependence of echo intensity on signal length must 
be investigated. 

Consider a curved surface, such as a sphere or an 
ellipsoid, each part of which reflects sound specularly 
as a mirror would. This surface is normal to the inci¬ 
dent beam at only one point, and only one ray is re¬ 
flected back to the projector in the direction of the 
incident ray. Therefore, the echo intensity — and 
consequently the target strength — will be inde¬ 
pendent of signal length, and the echo structure will 
accurately reproduce the signal structure, unless mul¬ 
tiple transmission paths which result from surface- 
reflected or bottom-reflected sound, for example, 
give rise to multiple echoes. This result, derived on 
the basis of ray acoustics, is not valid if very short 
pulses are used, since the wave character of sound 
must then be considered. However, this result is 
correct if the pulse is at least several wavelengths 
long. 

On the other hand, consider an extended rough 
surface, each part of which reflects sound in all direc¬ 
tions. A pulse r seconds long is sent out from a 


projector a distance r from the target which has an 
extension z in the direction of the incident beam, 
as illustrated in Figure 4. Now the first part of the 
signal will reach the nearest part of the target at a 
time r/c after it was emitted where c is the velocity 
of sound and will be returned to the projector at a 
time 2 r/c. The last part of the signal will leave the 
projector at a time r, reaching the nearest part of 
the target at r + r/c and the farthest part of the 
target at a time r -(- r/c + z/c; it will return to the 
projector at a time t + 2 r/c -J- 2 z/c. The duration of 
the echo will be the difference between the time when 
the first part of the signal reaches the nearest part of 
the target and is returned, and the time when the end 
of the signal is reflected from the farthest part of the 
target and is received at the projector. Then if the 
duration of the echo is <r, 

a = t -(- 2 z/c. (20) 

19.3.1 Long Pulses 

First, let the signal be long compared to the ex¬ 
tension of the target (Figure 4A). Then 

<r ~ t (21) 

and the echo length will approximately equal the 
signal length. Assume that the reflected energy is 
always directly proportional to the incident energy, 
and therefore to the product of the signal intensity 
and the signal length. Then the echo intensity will 
depend on the signal intensity but not on the signal 
length. 

Now let the signal length equal the depth of the 
target in the direction of the beam (Figure 4B). Then 

2 z 

cr = r H-= 3r, (22) 

c 

and the echo length will be three times the signal 
length. The echo will no longer resemble the signal, 
as the echo intensity grows to a maximum when the 
target is illuminated by the entire signal. 

19.3.2 Short Pulses 

Lastly, let the signal be short compared to the 
depth of the target (Figure 4C). Then 



and the echo length will approximately equal twice 
the extension of the target in the direction of the 
beam. The echo intensity now will depend on the 




WAVE CHARACTER OF SOUND 


351 


signal length as well as the signal intensity, since the 
reflected energy will be less for a short signal than a 
long signal and therefore — as long as the echo length 
remains constant — the echo intensity will be re¬ 
duced. 

For short pulses, then, the echo intensity and 
therefore the target strength will depend on the pulse 
length. In practice, however, fluctuations in the 
course of each echo, and from echo to echo, tend to 
obscure this relationship for any individual echo. For 
long pulses, the echo very closely reproduces the 
signal envelope, which is usually square-topped. For 
short pulses, however, fluctuations in echo intensity 
result in a very irregular hashed structure where a 
sharp peak or group of peaks stands out clearly 
against a background which is sometimes 10 db lower. 

The peak echo intensity, which is usually used in 
computing target strengths, is, in general, different 
from the average echo intensity. Therefore, for short 
pulses the peak echo intensity may be considerably 
different from the average echo intensity and may 
vary in a different way with signal length. Peak and 
average echo intensities, and how they vary with 
signal length, are discussed in Sections 21.6.4 and 
23.5.1. 

19.4 WAVE CHARACTER OF SOUND 

Ray acoustics has been used exclusively through¬ 
out this chapter in defining target strength, in de¬ 
riving the target strength of a sphere, and in dis¬ 
cussing target strength as a function of experimental 
variables, just as ray acoustics was employed in 


Chapter 3 of this volume in treating the transmission 
of sound through sea water. Experience shows, how¬ 
ever, that sound does not always travel in straight 
lines, and that, for many purposes, ray acoustics is 
inadequate in explaining and interpreting underwater 
sound phenomena. An alternative approach in terms 
of wave acoustics becomes necessary. 

Throughout this chapter it has been tacitly as¬ 
sumed that sound is propagated along straight lines 
as sound rays, and that reflection is wholly specular, 
in other words, that the angle of reflection always 
equals the angle of incidence. Many modifications 
must be introduced if allowance is made for the 
various wave phenomena affecting echo ranging. 
Sound is diffracted when it strikes a target or parts 
of a target whose dimensions approximate its wave 
length. Thus, the previous discussion applies only to 
targets considerably larger than the wave length. 
For the same reason, these results apply only to 
pulses whose length in the water is at least several 
wave lengths. In addition, sound reflected from one 
part of a target may interfere with sound reflected 
from other parts. Much of the fluctuation commonly 
encountered in analyzing echoes from underwater 
targets is attributable to interference. The results in 
the preceding section on echoes from extended tar¬ 
gets are valid only if the interference effects arising 
from constructive or destructive interference can be 
eliminated by averaging over several successive 
echoes. The effect of the wave length of sound on 
target strength for both specular and nonspecular 
reflection is discussed in greater detail in Sections 
20.4 and 20.6. 



Chapter 20 


THEORY 


I n chapter 19 the concept of target strength was 
introduced and its meaning defined quantita¬ 
tively; then the target strength of a perfectly reflect¬ 
ing smooth sphere was derived in terms of ray 
acoustics. The theoretical background will be pre¬ 
sented in this chapter in terms of wave phenomena 
with a mathematical discussion of the reflection of a 
sound wave from a target of any shape, and a review 
of the early theoretical calculations of the target 
strength of a submarine. 

In principle, the reflection of sound from a target 
can be exactly determined by solving the wave equa¬ 
tion derived in Chapter 2 of Part I, as long as the 
proper boundary conditions at the surface of the 
target are satisfied. In practice, an exact computation 
along these lines is mathematically very difficult; the 
difficulties are most marked for targets large com¬ 
pared to the wavelength of the incident sound. Even 
for a sphere the rigorous analysis which has been 
worked out *• 2 is rather complicated. Numerical ap¬ 
plications of these precise formulas have been pub¬ 
lished 3 for a rigid sphere, whose circumference is 
from 1 to 10 times the wavelength; the results pro¬ 
vide an interesting example of the exact behavior of 
reflected sound in one simple case. However, even 
for such relatively small targets the mathematical 
analysis becomes tedious. 

20.1 APPROXIMATIONS 

To obtain more general results, various approxima¬ 
tions must be made, physical as well as mathematical. 
The mathematical assumptions made in this chapter 
are fairly standard and are believed to give essentially 
correct results. The physical assumptions about the 
nature of the reflecting surface are more important, 
however, and require some justification. 

In the first place, most of this chapter applies only 
to targets which are large compared to the wave¬ 
length of the sound, and whose surface is smooth; in 
other words, the radius of curvature of the surface is 


also large compared to the wavelength. These re¬ 
strictions seem legitimate for most targets of practical 
interest in echo ranging. 

In the second place, the material of which the tar¬ 
get is composed is assumed to be rigid. In terms of 
sound reflection, a target is said to be rigid if p\C\f tnti 
is negligibly small, where pi and C\ are the density and 
sound velocity in the surrounding medium, and p-i and 
C 2 are the density and sound velocity in the target. 
When this condition is not fulfilled the problem be¬ 
comes much more complicated. In most cases of in¬ 
terest to subsurface warfare, the target is bounded 
by thin metal plates, inside which there may be air 
or w'ater. The reflection of sound from such plates has 
been studied, 4,5 and the results obtained show" that 
even for plates only 34 in- thick, such as generally 
constitute submarine superstructures, the reflection 
is practically perfect; transmission and absorption 
are negligible. Thus, the assumption of perfect re¬ 
flection from practical targets seems justified. 

Some of the additional assumptions w r hich may be 
made in the discussion of the reflection from targets 
are discussed in an early British report. 6 This work is 
particularly interesting because it presents the most 
complete available application of theory to the target 
strength of underwater objects. 

The essential elements of the theory of target 
strength, restricted by the physical assumptions 
which have been made here, are presented in the 
following sections. First, an approximate but general 
formula is derived for the pressure of the sound re¬ 
flected from a target. In Section 20.3, this result is 
further simplified to give an equation for the target 
strength of a reflecting surface in terms of the so- 
called Fresnel zone theory. This latter equation is 
then used to find practical formulas for the target 
strengths of simple geometrical shapes, vdiich are 
applicable to the major reflecting properties of sub¬ 
marines; the application to an actual submarine is 
described in Section 20.5. All this latter analysis ap¬ 
plies only to long pulses. The last two sections are 


352 


REFLECTED PRESSURE 


353 


devoted to a qualitative discussion of the reflection 
from targets small compared to the wavelength, and 
the echoes obtained with very short pulses. 


20.2 


REFLECTED PRESSURE 


Consider first a sound beam striking a surface ele¬ 
ment dS of a perfectly reflecting, smooth and rigid 
underwater target. Since this surface is rigid, the 
primary effect of the target is to prevent the water 
from moving perpendicularly to dS at the surface of 
the target. In other words, at the surface of the tar¬ 
get, the velocity u of the water, measured along a line 
perpendicular to the surface, must be zero, or 

u, = 0, (1) 

where the z axis, at the point of incidence, is perpen¬ 
dicular to the target surface. By differentiating equa¬ 
tion (1) with respect to the time t, 


— = 0. 
dt 


( 2 ) 


But from equation (17) in Chapter 2 of this volume, 
du z dp 
P dt dz 


(3) 


where p is the density of the medium, p the pressure 
of the sound wave, and z the coordinate perpendicular 
to the surface. 


20 . 2.1 Boundary Condition 

Substitution of equation (3) into equation (2) gives 



which means that for a rigid target the component of 
the pressure gradient perpendicular to the surface 
must vanish at the surface. This is the boundary con¬ 
dition which the solution of the wave equation must 
satisfy. 

In the absence of the target, the sound source will 
send out a wave whose resulting pressure at any par¬ 
ticular point may be denoted by pi; then p\ must be a 
solution of the wave equation [equation (27) in Chap¬ 
ter 2.] In the presence of the target, this pressure pi 
does not satisfy the resulting boundary conditions at 
the surface of the target. The actual sound pressure p, 
which must satisfy both the wave equation and the 
boundary conditions at the target surface, may be 
written as 

(5) 


where p* constitutes the correction which must be 
added to the undisturbed sound pressure pi in order 
to satisfy the boundary conditions at the surface 
of the target. 

By differentiating equation (5) with respect to z 
and by substituting the result into equation (4) 


dpi dp 2 
dz + dz ’ 


( 6 ) 


which is another way of expressing the boundary 
condition. 

Because the wave equation is a linear homogeneous 
differential equation, the difference between the two 
solutions p and pi is again a solution, and p 2 by itself 
must therefore satisfy the wave equation. In other 
words, the total sound field may be interpreted as the 
combination of two sound fields. One of these, whose 
pressure at any specified point is p u is called the 
incident sound; the other, whose pressure at the same 
point is p 2 , is the reflected sound. Each of these 
quantities satisfies the wave equation, but only their 
sum satisfies the boundary conditions at the target. 
In some places, the measured sound pressure may oc¬ 
casionally consist wholly of one or the other of these 
two sound fields, depending on whether only the 
sound projected from the source, or only the sound 
reflected from the target is measured. The problem 
tackled in this chapter is the evaluation of the re¬ 
flected sound alone, since it is this quantity which is 
most important in echo ranging. Therefore, an ex¬ 
pression for p 2 must be derived. 


20 .2.2 Mathematical Formulation 


To obtain, rigorously, a general expression for p 2 is 
usually a very difficult problem. It is comparatively 
easier to obtain an approximate solution by distrib¬ 
uting, over the surface of the target, point sources of 
sound. Then, if the distribution and strengths of 
these point sources over the area are correctly chosen, 
these point sources will emit sound in such a way as 
to cancel the pressure gradient of the wave incident 
on the surface, thus satisfying the condition (6). 

For a single point source, the solution of the wave 
equation is 


p = 

r 


(7) 


where p is the pressure of the sound field at a distance 
r from the source, B is a constant which measures the 
strength of the point source, i is\/—1> t is the time, 


p = Pi + P2, 



354 


THEORY 


/ is the frequency and X the wavelength of the sound. 
If the reflecting target itself is considered to be made 
up of many point sources distributed over its surface, 
p becomes P 2 , the reflected pressure; and the pressure 
dpi produced by all the point sources located in a 
surface element dS is 

dp 2 = -e 2ri(/t ~ r/x) dS, (8) 

r 

or the pressure p 2 produced by the entire target be¬ 
comes 

p 2 = J ^e 2ri{ft ~ rM dS (9) 

s 

where G is essentially the average value of B in equa¬ 
tion (7) for each individual source multiplied by the 
number of sources per unit area; the integral is eval¬ 
uated over the entire area S. 


z 



Figure 1. Transformation to polar coordinates. 


This quantity G is a measure of the number and 
strength of the point sources over the area; in general 
G will vary over the target surface. The function G 
must be chosen so that the resulting sound pressure 
p 2 satisfies the boundary condition (6) on the surface 
of the target . 

The value of G at a particular point of the target 
surface will be assumed to be completely determined 
by the incident sound pressure at that point. This 


assumption is not rigorously correct, but it leads to a 
good approximation if the target has a surface whose 
radius of curvature is everywhere large compared 
with the wavelength. 

First, a relationship between the value of G at any 
point and the resulting gradient of p 2 at that point 
will be derived. Then, the gradient of p 2 may be re¬ 
placed by minus the gradient of pi, because of the 
boundary condition (6). In this manner, a direct 
relationship will be obtained between the incident 
sound field p x on the target surface, and the value of 
G required to compensate the gradient of pi. 

Because of the assumption made that the gradient 
of p 2 at the point on the target surface is determined 
primarily by the value of G at that point, a particu¬ 
larly simple model may be considered and the result 
generalized. The pressure gradient at the center of a 
disk illustrated in Figure 1 will be derived, on the as¬ 
sumption that G is constant over the surface; in other 
words, the density of point sources on the surface of 
the disk is assumed to be uniform. If polar coordi¬ 
nates p and 6 are introduced, the integral (9) for the 
pressure on the z axis can be transformed as follows: 

p, = 2*G f" e*'‘ U ‘~ rm — • 

Jp =o r 

or, since r 2 = p 2 + z 2 , 

X Vfli+is 

r/ » dr (1Q) 

Equation (10) may be integrated directly and gives 
for the sound pressure on the z axis 

a« 

and by differentiating p 2 with respect to z, the gradi¬ 
ent at p 2 perpendicular to the surface becomes 

?■«[ 

( 12 ) 

For the point on the surface where z = 0, the gradient 
reduces to 

(f )„ 0 = - (13) 

which is independent of the radius R of the circular 
surface. This result confirms the assumption that the 
gradient of p 2 at any point on the surface is deter¬ 
mined only by the value of G in the immediate 
vicinity of that point; thus G is independent of possi¬ 
ble variations in G at other points. Actually it is 


Vr 2 + z 2 


_ g2rt(/«-zA)J 









REFLECTED PRESSURE 


355 


rigorously correct only for a plane surface, but results 
in a good approximation for other surfaces as long as 
the curvature is small over the distance of one wave¬ 
length. 

Consequently, it will be assumed that in general 
the gradient of p 2 and the value of G are related to 
each other at each point on the target surface by the 
equation 


n - 

2tt dz ' 


(14) 


If the boundary condition (6) is to be satisfied, 
— dp 2 / dz in equation (14) may be replaced by 
dpi/ dz, and in terms of the incident-sound wave 


r - * - 2 «/idpi 
2ir dz 


(15) 


Since the incident sound pressure is usually a har¬ 
monic wave, it may be locally described by 

pi = be 2ri(ft ~ qM (16) 

where b is the amplitude of the wave, / its frequency 
and X its wavelength, and q a coordinate parallel to 
the direction of propagation. The derivative of p x in 
the direction of propagation is then 

^ = A). (17) 

dq X 

The derivative of the amplitude b has been neglected 
in this equation since this derivative is usually negli¬ 
gible at distances from the source of many wave¬ 
lengths. 

In any other direction, the derivative will equal 
expression (17) multiplied by the cosine of the angle 
between the direction chosen and the direction of 
propagation q. If the angle between the direction of 
propagation of the incident sound wave and a line 
perpendicular to the target surface is 6, then the 
derivative of p x along a line perpendicular to the 
target surface is 

^ = -—b cos d e 2 ^ ft - qM • (18) 

dz X 

If this expression is substituted in equation (15), G 
becomes 

G = -4cos0<T 2 ™ 9A - (19) 

X 


It is particularly interesting to evaluate the wave 
amplitude b for the case where the incident wave is 
caused by a point source of sound at a point P, a 
distance r' from the point of the target surface con¬ 
sidered. If at unit distance from P the amplitude of 


the incident spherical wave is B, then the local ampli¬ 
tude b equals B/r'; the coordinate q may be replaced 
by r', and equation (19) assumes the form 

G = -"Cosfl<r WA - (20) 

X r 

If this expression for G is substituted into equation 
(9), the resulting integral for the reflected sound pres¬ 
sure p 2 becomes 



where B is the amplitude of the original point source 
at unit distance, r' is the range from the source to a 
point on the surface of the target, r the range from 
that point on the target surface to the point in space 
where p 2 is to be found, / the frequency and X the 
wavelength of the sound. The integration is to be 
carried out over the whole target surface S of which 
dS is a surface element. 

20.2.3 Physical Interpretation 

So far the discussion has been wholly mathematical, 
without the benefit of a physical argument to support 
and justify the approximations made. Physically, the 
analysis is based on the fundamental principle that 
in the vicinity of a rigid surface the fluid motion in a 
direction perpendicular to that surface must vanish. 
If the incident pressure wave made the fluid move so 
as to violate this condition, the rigid surface would 
exert a force on the adjacent fluid elements just can¬ 
celing this motion perpendicular to the surface. This 
effect may be imagined by replacing each element of 
area on the target surface by a small piston capable 
of moving in a direction perpendicular to the surface. 
In the absence of the boundary condition, each of 
these pistons would be moved back and forth in 
rhythm with the motion of the adjacent fluid element. 
In order to act as parts of a rigid surface, however, 
these little pistons must each be pushed by a force 
opposite to that of the motion of the fluid, just 
sufficient to keep each piston permanently balanced 
in its original position. This alternating force which 
each piston exerts on the fluid has the same net effect 
as the force which a transducer exerts on the sur¬ 
rounding fluid, in other words, each acts as a sound 
source with spherical wavelets emanating from each 
individual piston. The appropriate amplitude and 
phase of these wavelets has been calculated above. 
The total reflected sound field then represents the 
superposed effects of all these individual wavelets. 



356 


THEORY 


20.3 FRESNEL ZONES 


In this section, equation (21) will be applied to 
compute a general formula for the target strength of 
a smooth and rigid target. Here, smooth means that 
the radius of curvature of the target surface is large 
compared to the wavelength of the sound striking it. 
Moreover, the target is assumed to have a relatively 
simple shape, always convex, with no marked bumps 
or protuberances. While this ideal target hardly re¬ 
sembles most actual targets, the consideration of this 
simple problem gives some insight into the means by 
which sound waves are actually reflected. Even under 
these special assumptions, however, it will be shown 
that the integral in equation (21) can be readily 
evaluated only by an additional approximation, first 
suggested by the French physicist, Fresnel. 

Consider only the case where the echo is observed 
back at the sound source; this case corresponds to 
the situation of chief practical interest, as pointed out 
in Section 19.1, and in addition simplifies the com¬ 
putations. Then r = r' and equation (2l) reduces to 


Th 


iBe 2nft r cos 

X J ~r* 


—4ir ir/X 


dS. 


( 22 ) 


In the integral, however, both 6 and r vary over the 
surface of the target. Therefore, the integral cannot 
be evaluated by elementary methods, except for cer¬ 
tain special cases illustrated in Section 20.4 where 
an exact integration can be carried out. For most 
practical purposes, however, an expression for the 
reflected sound pressure p 2 and, therefore, for the 
target strength T can be derived by means of an 
approximate method, which was originally developed 
in optics and which is known as the method of 
Fresnel zones. 

This method is based on the mathematical anal¬ 
ysis developed in the preceding section. Physically, 
according to equation (21), every point on the target 
surface which is struck by the incident sound pres¬ 
sure wave becomes in turn a center of outgoing wave¬ 
lets so that the points on the target surface may be 
considered “secondary sources” of sound. In optics, 
this is called Huyghens’ principle. Simple addition of 
the sound pressure in each individual wavelet will 
give the reflected sound pressure p 2 . 

Now, every wavelet has a phase depending on the 
total distance traveled by the sound out to the target 
and back. In general, these wavelets interfere both 
constructively and destructively. Destructive inter¬ 
ference leads to cancellations due to the phase dif¬ 


ferences. But a sharp maximum of amplitude —- due 
to constructive interference, where wavelets whose 
amplitudes are all of the same sign are superimposed 
— exists in the direction corresponding to specular 
reflection. This is the direction in which the beam is 
reflected according to ray acoustics. A quantitative 
calculation of the amplitudes of the different wave¬ 
lets will show exactly how much energy is reflected 
in different directions. In this way, wave acoustics 
can be shown to give the same results as ray acoustics 
when the wavelength is very short. 

20 . 3.1 Method 

To compute the amplitudes and phases of the 
different wavelets, the surface may be divided into 
successive areas from which all the wavelets emitted 
are approximately in phase and thus do not interfere 
destructively. This is the Fresnel method. According 
to this method, consider a series of wave fronts pro¬ 
ceeding outward from a source at the point P, 
separated by a distance X/4 from each other, where 
X is the wavelength of the projected sound. When 
they strike the target, the surface of the target is 
intersected by these wave fronts in a series of curves 
which divide the surface into the so-called Fresnel 
zones. The phase of each reflected wavelet, measured 
back at P, is 2 irft — 4irr/\. Since ir equals the product 
of 47r/X times X/4, the distance between two adjacent 
zones, the wavelets from each zone have an average 
phase difference of t from the Avavelets of the ad¬ 
jacent zones. But a change of phase by the amount ir 
results in multiplication of the amplitude by — 1; 
hence the wavelets from each zone interfere destruc¬ 
tively with those from the two adjacent zones. The 
advantage of the Fresnel-zone approach, as will be 
shown, is that most of the zones cancel each other, 
leaving only the effects of the first and last zones to 
be considered. 

While the analysis can be carried out for the Fres¬ 
nel zones defined by the wave fronts at any one time, 
it is simplest to take the zones resulting when one of 
the particular wave fronts considered is just tangent 
to the closest point on the target. Let R be the value 
of r at this point, in other words, let R be the distance 
from the sound source and receiver at P to the nearest 
point on the target. The first zone is the area on the 
surface of the target intercepted by the wave front 
which is a distance X/4 from the wave front tangent 
to the target. In general, the position of the nth zone 
is then determined by the inequality of equation (23) 




FRESNEL ZONES 


357 


R + (n - 1)- < r < R + n; (23) 

4 4 

where r is the distance from the source to any point 
in the zone. 

It S n is the area of the /ith zone, equation (22) can 
be written as a sum of integrals in which each integral 
extends over only one zone. Then 


P2 



®E f — 

n J Sn r 2 


— 47n>/X 


dS; 


(24) 


the sum, denoted by the symbol 2, extends over as 
many values of n as there are zones. To evaluate the 
integral in equation (24), define a new variable u„ for 
the nth zotie by 

r — R — (n — 1) X/4 


X/4 


(25) 


If this equation is substituted in inequality (23), u n 
satisfies the relationship 


0 < u n < 1. (26) 

Thus, u n increases from 0 at the near side of the 
n zone to 1 at the far side. Equation (24) then be¬ 
comes 

p 2 = —e-™"dS. (27) 

X n J Sn f 2 


Since e”” = - 1, then e ~ (n ~ 1)ni = (-1)* -1 , and the 
reflected sound pressure p 2 may be written 

p 2 = e 2 " (/< “ 2R/x ) [ P 1 - P 2 + P 3 • • • 

X 

+ (_l)^- 1 P n ], (28) 

where N is the total number of zones and P n is 
defined by 

P„ = f —e~ Hun dS. (29) 

J Sn r 2 

In each integral u n lies between 0 and 1 and obeys 
inequality (26). 

For targets large compared to the wavelength, 
whose surface is not too sharply curved, there will be 
a large number of zones and the values of P„ found 
in successive zones will not change very rapidly as n 
is changed. The quantity r 2 will scarcely change at 
all if the distance to the sound source and receiver 
is much greater than the size of the target. The 
quantity cos 9 may decrease from 1 in the first zone to 
a small value for the higher zones, but if the target is 
much larger than the wavelength and if its surface is 
not curving too sharply, the change in cos 9 from one 
zone to the next will not be large. Similarly the area 
S n of successive zones will not change very rapidly. 


The factor e~ Tlu " varies in the same way in all the 
zones. Thus, on the average, the partial pressure P n 
of the wavelets reflected from the nth zone may be 
assumed to be equal to the average of the correspond¬ 
ing partial pressure of the wavelets for the preceding 
and following zones, or 

Pn = i(P„-1 + P„+1). (30) 

Equation (30) forms the basis of the Fresnel approxi¬ 
mation. It may be expected to become increasingly 
accurate for a smooth surface as the wavelength X 
decreases indefinitely and the order number n of the 
particular zone increases indefinitely. 

With this approximation, the successive terms in 
equation (28) cancel out, and the sum of all the P„’s 
in equation (28) becomes simply 

A - P 2 + P 3 • • • + (-1 ) N P n 

= 4[Pi + (— 1 )' v PjvQ. (31) 
In most practical cases, the value of cos 9 for the 
last or Nth zone is zero, since the target surface at this 
point is tangent to the sound rays. Thus P N vanishes 
and the sum of P„ over all the zones is simply one- 
half the value of P for the first zone. This is a particu¬ 
larly interesting and important result ; if only half of 
the first zone participates in the reflection, and the 
entire target surface beyond it is neglected, the re¬ 
flected sound wave is the same as if the entire target 
were regarded as the reflecting surface. Only a small 
part of the target surface perpendicular to the sound 
rays produces the entire reflection; the reflection from 
this small region is sometimes called a “highlight” as 
it is in optics. It is this result, derived on the basis 
of wave acoustics, which corresponds to the specular 
reflection based on ray acoustics in Chapter 19. 

Therefore, set Pjy equal to zero in equation (31), 
substitute the result into equation (28), and p 2 be¬ 
comes 

V2 = -^‘(/'-Mp,. (32) 

Now, since the value of r will be almost constant 
throughout the first zone, unless the source is only 
a few wavelengths away, r may be replaced by R, the 
shortest distance from the source to the target. 
Equation (29) may therefore be written as 

= 1 I cos 9 e~* iu 'dS. (33) 

R 2 Js l 

Finally, by combining equations (32) and (33) the 
pressure p 2 in the reflected wave becomes 

p 2 = f cos 9 e~ wiui dS. (34) 

2 \R- Js. 




358 


THEORY 


20 . 3.2 Application 


To find the target strength corresponding to the 
pressure of the reflected wave in equation (34), equa¬ 
tion (6) in Chapter 19 may be written in the form 
T = 20 log |p 2 | - 20 log |B| + 40 log R (35) 


where the vertical bars mean that absolute values of 
the complex quantities involved must be taken. The 
term 20 log |p 2 | is the rms echo level E where p 2 is the 
actual echo pressure; 20 log |B[ is the rms source 
level S at 1 yd; and 40 log R is twice the transmission 
loss from 1 yd out to the target at a range R, ex¬ 
cluding attenuation losses. Strictly speaking, the rms 
level is the average value of the square of the real 
part of the complex quantity rather than the abso¬ 
lute value; however, a more elaborate computation 
along these lines leads to exactly equation (35). If 
equation (34) is substituted into equation (35) the 
target strength T becomes 


r = 20 log 


1 f 

— | cos 6 e 
2\ Js, 


"dS 


(30) 


where the bars again denote that an absolute value 
must be taken. The quantity U\ in the exponent is 
defined by 

Mi = ^(r - R). (37) 

A 


Si in equation (30) is the area of the target in which 
U\ is less than 1; the integral is evaluated only over 
those surface elements lying within Si. 

The evaluation of equation (30) provides the solu¬ 
tion of the problem presented at the beginning of this 
section. 


20.4 TARGET STRENGTH OF SIMPLE 
TARGETS 

In this section, equation (30) will be used to com¬ 
pute the target strength of relatively simple surfaces, 
such as spheres, cylinders, and other objects, which 
have a single highlight. The results obtained may 
also be applied to more complicated surfaces, as long 
as the radius of curvature is greater than the wave¬ 
length. Whenever several highlights are present, the 
reflected wave is the sum of the waves reflected from 
each one separately. In general, they will interfere. 
However, if an average is taken over a considerable 
spread of target aspects, and if the highlights are 
spaced much further apart than the wavelength, the 
interference will tend to be random; in this situation, 
the intensity of the echo is simply the sum of the 
intensities computed for each highlight individually. 


20 .4.1 Sphere 

The target strength of a sphere, on the basis of 
wave acoustics, may be easily derived from equation 
(36). The results of this analysis may be used not 
only for a perfect sphere but also for any target sur¬ 
face whose first Fresnel zone is essentially spherical. 

Consider a wave from a source P striking a sphere 



of radius A, illustrated in Figure 2, whose nearest 
point is a distance R away from the source. If <f> is the 
angle subtended at the center of the sphere by Q, 
which bounds an element dS of area, dS is simply 

dS = 2 tA~ sin 4>d4>. (38) 

By the law of cosines, the distance r from the source 
P to the point Q is given by 

r 2 = (R + A) 2 -(- A 2 - 2 A(R + A) cos <f> 

= R- + 4A (A + R) sin'Y (39) 


When R is much greater than A, r is approximately 

(40) 


„ , 2A(A + R) . 
«+----:u - 


The quantity iii from equation (37) is then 
8A (A + R) . .,</> 

" - XS ' SU ’- 2’ (41) 

For short wavelengths, sin <j>/2 will be very small in 
the first zone and may be set equal to <f>/2 ; similarly 
cos 6 in equation (36) may be replaced by one. There¬ 
fore, if equations (38) and (41) are substituted into 
equation (36), the target strength of a sphere becomes 

i r 0 ° 

T = 20 log — e~ wlx *'2ir A 2 <Ar/</>, 

2\Jo 


(42) 


where 


and 


2A (A + R) 
\R 


T0o = I- 


(43) 


(44) 


The integration may be carried out and yields 











TARGET STRENGTH OF SIMPLE TARGETS 


359 


f 

n 


-nix# 




— 2irix 


irix 


Thus equation (42) becomes 

T = 201og^-\ 
x\ 


(45) 


(46) 


and if equation (43) is substituted for x, the target 
strength reduces to 

T ~ 20 l0g 2(l A/fi)' < 47 > 


This expression is valid only when the distance from 
the source to the sphere is at least several times 
greater than the sphere diameter. When R is very 
much greater than A, equation (47) simply becomes 


T = 20 log-, 


(48) 


which is identical to equation (10) in Chapter 19 
derived on the basis of ray acoustics. At shorter 
ranges, equation (36) is still applicable, but must be 
evaluated more accurately. It may be noted that the 
value of T in equation (47) is based on the assump¬ 
tion that the transmission loss to the nearest point 
of the sphere is used in equation (35). If the trans¬ 
mission loss to the center of the sphere is used in¬ 
stead, T must be increased by 40 log (1 + A/R) and 
increases as the range becomes shorter. 

As already pointed out, equation (47) may be ap¬ 
plied whenever the first Fresnel zone of a reflecting 
surface is spherical in shape, and has a radius of 
curvature A much larger than the wavelength X. The 
result is independent of the wavelength. Equation 
(36) could be evaluated more accurately to find a 
dependence of T on wavelength. This dependence 
would be appreciable only when the wavelength was 
no longer much smaller than the sphere radius A, in 
which case the total number of Fresnel zones would 
no longer be large. Since the accuracy of the Fresnel 
method is doubtful under these conditions, the wave¬ 
length dependence found in this way would not be 
very reliable unless confirmed by a much more 
elaborate investigation. 


20.4.2 General Convex Surface 

More generally, the curvature of a surface cannot 
be described by a simple single radius of curvature. 
In such a case, the boundary of the first Fresnel zone 
will not be a circle, as was the case for a spherical 
surface. In a more general case, this boundary will be 


elliptical in shape, and the surface intersected will 
have two principal radii of curvature Ai and A 2 , 
which will usually differ from point to point. 

These radii may be defined as follows. Let 0 be a 
particular point on the surface and let OC be a line 
perpendicular to the surface at the point 0. Any 
plane containing OC will intersect the surface in some 



Figure 3. Reflection from any convex surface. 


line QOQ', as in Figure 3. In the neighborhood of the 
point 0 this curve is approximately a circle of radius 
A. However, as the plane intersecting the target is 
rotated about the line OC, the radius A of the curve 
QOQ' will vary. It will have a maximum value A x and 
a minimum value A 2 , in general, as the plane rotates 
through 180 degrees. Furthermore, according to dif¬ 
ferential geometry, these two radii will be 90 degrees 
apart. These two quantities Ai and A 2 are called the 
principal radii of curvature of the surface at the point 
O. If they do not change rapidly with position on the 
target surface — more particularly, if they are ap¬ 
proximately constant at all points in the first Fresnel 
zone — the target strength of the surface may be 
computed. 

The derivation is more complicated than that in 
Section 20.3.1 and will not be given here. The result 
of the analysis is in the following equation 








THEORY 


360 


T = 10 log - 


AiAi 


2(1 + A y /R)(l + A a /R)’ 


(49) 


which reduces immediately to equation (47) when 
Ai is equal to A 2 . While equation (47) was valid only 
if Ai/R was moderately small, equation (49) is ap¬ 
plicable even if A\/R is very large as long as A 2 /R 
is still small. Equation (49) cannot be used, however, 
when either Ai or A 2 approaches the wavelength of 
sound. 


20.4.3 Cylinder 

For an infinitely long cylinder, equation (49) may 
be applied directly by letting one radius of curvature 
Ai be infinite. The target strength found for this case 
reduces to 

r=ioi ° g Ki+£/*)’ (5o> 

where A 2 is the radius of curvature of the cylinder. 
This equation is valid only when the wavelength of 
the sound is much less than the radius of curvature 
of the cylinder, and when this radius in turn is much 
less than the range. 

For an actual cylinder equation (50) may be used 
only if the cylinder is perpendicular to the sound 
beam at some point, and if the cylinder is long enough 
to include at least the first few Fresnel zones. The 
expression may therefore be used only at moderate 
ranges, since with increasing range the length of the 
first Fresnel zone increases infinitely. 

To compute the range beyond which equation (50) 
cannot be used, let the length of the cylinder be L, 
and let the sound source lie in a plane which is per¬ 
pendicular to the axis of the cylinder and bisects the 
cylinder. Then the path length r to the end of the 
cylinder is 



The length of the cylinder will include many Fresnel 
zones if r given by equation (51) exceeds R by many 
wavelengths. Therefore equation (50) may be used 
only as 

R<<\- (52) 

A 

For example, for a wavelength of 4 in., corresponding 
to a frequency of about 15 kc and a cylinder 10 ft 
long, the range must be much less than 100 yd if 
equation (41) is to be used. 

At long ranges, R is much greater than L 2 /X, and 
the computed length of the first Fresnel zone exceeds 


the length of the cylinder. In this case, instead of 
using the approximation (30) an exact integration of 
equation (22) over all the zones is possible, provided 
the variation of cos 6 is neglected, and the target is 
far away from the source. Thus in equation (36), in¬ 
stead of one-half the integral over the first zone, we 
may take the same integral over all the zones. With 
the same approximation for u made in the previous 
section, the target strength becomes 


T = 10 log 


L 

2 


A 

2A(1 +A/R) 


(53) 


At these longer ranges, the target strength is again 
independent of the range, in agreement with the 
comments made in Chapter 19. However, equation 
(53) presents one case in w r hich the target strength 
varies appreciably with changing wavelength, even 
when the wavelength is much smaller than the target. 

For intermediate values of the length of the cylin¬ 
der, both the first and last zones must be considered. 
A more exact evaluation of equation (22) can be car¬ 
ried through in this special case by use of particular 
functions called Fresnel integrals, which have been 
tabulated. 


20.4.4 Reflection at Close Ranges 


The formulas developed so far in this chapter are 
applicable to many simple shapes provided the sound 
source and receiver are not too close to the target. 
The target strength at close ranges may also be found 
directly from equation (36). Detailed results have 
been worked out for cases of this nature, but will not 
be reproduced here. In general, when R becomes 
much less than the principal radii Ai and A 2 , the 
reflection can best be described as reflection from a 
plane surface. In the limiting case where R/A 2 is 
negligible, 


P2 2 R 6 


(54) 


as long as the sound field this close to the target obeys 
the inverse square law; for a large directional trans¬ 
ducer, this condition is not likely to be satisfied at 
very close ranges. If equations (35) and (54) are 
combined, the target strength becomes 

T = 20 log | • (55) 

Formulas for the target strength of various types of 
objects, such as two cones placed base to base, and 
a circular disk placed at an angle to the sound beam, 
are given in reference 6. 







NONSPECULAR REFLECTION 


361 


20.5 REFLECTIONS FROM SUBMARINES 

Expressions were developed in the preceding sec¬ 
tion for the target strengths of various surfaces in 
terms of the reflected pressures. These formulas were 
employed in a theoretical study 7 in order to calculate 
mathematically the target strength of a German 
submarine. 

From an examination of blueprints of the U570, 8 a 
517-ton German U-boat captured by the British 
early in the w r ar and renamed HMS/M Graph, the 
radius of curvature of the surface of the hull and 
conning tower at different points was obtained. In 
computing the results, the submarine was approxi¬ 
mated by an ellipsoid of revolution, whose semi-axes 
were 110 and 7 ft. The results were then corrected 
for the reflections from the conning tower, which was 
assumed to be a cylinder with a “tear-drop” cross 
section. 

Target strengths were found from equations (49) 
through (53) in terms of the range and the radii of 
curvature for different submarine cross sections; 
ranges of 8, 12, 16, 200, and 1,000 yd were used. At 
ranges where the conning tower did not include a 
large number of zones, the Fresnel integrals [ob¬ 
tained when equation (22) is integrated exactly 
along the length of a cylinder] were used. The 
calculations were actually carried out in terms of 
reflection coefficients, which differ somewhat from 
target strengths derived in this chapter. The results 
of these computations are presented in Chapter 23 
together with the results of the direct and indirect 
measurements. 

20.6 NONSPECULAR REFLECTION 

So far only reflections from highlights on a target 
surface have been discussed. These highlights cor¬ 
respond to specular reflections in optics and give 
much the same predictions as those found from the 
ray theory. In particular, the echo is assumed to 
come only from that region of the target where 
the surface is nearly perpendicular to the incident 
sound wave. This section discusses those cases 
where such reflection cannot occur and where the 
observations cannot be explained in this way. At 
the present time, however, different types of non- 
specular reflections have not been identified wflth 
any observed reflections from actual targets, so that 
at most this section can only suggest the theoretical 
expectations. 


20.6.1 Rough Surfaces 

The most simple type of nonspecular reflection is 
that from a rough surface, that is, a surface whose 
irregularities are much larger than the wavelength. 
Practical formulas applying to this kind of nonspecu¬ 
lar reflection from various underwater targets are de¬ 
rived in reference 6. The wavelength of sound is so 
much greater than that of light, however, that such 
reflections, which are common in optics, are not to be 
expected in underwater acoustics. The presence of 
bubbles on or near the surface of a target can, how¬ 
ever, give rise to a diffuse reflection with sound scat¬ 
tered in all directions; the reflection of sound from 
bubbles is described in detail both theoretically and 
experimentally in Chapters 26 through 35, which 
deal with the acoustic properties of wakes. 

20.6.2 Diffraction 

Another type of nonspecular reflection is that from 
a surface which has no highlights. Consider, for ex¬ 
ample, a smooth rigid plane surface in the form of a 
square, set at an angle relative to the incident rays. 
This surface will reflect sound specularly, but not 
back to the sound source. In addition, however, some 
sound will be reflected in other directions; some of it 
will be reflected directly backward. This phenome¬ 
non corresponds essentially to the diffracted sound 
observed when a wave passes through a square 
aperture, and the echo intensity will decrease as 
(y/X) 2 , where y is the length of the square and X is 
the wavelength. The Fresnel zone theory may again 
be applied, provided that the effects of both the first 
and last zones are considered. No results have been 
worked out along this line, however. 

20 . 6.3 Scattering 

A third type of nonspecular reflection is that from 
objects much smaller than the wavelength. The 
Fresnel zone theory is not applicable to such small 
targets, and even the basic equation (21) derived in 
Section 20.1 is no longer valid, since the derivation 
assumes that the radius of curvature of the surface is 
greater than the wavelength. Corresponding analyses 
have been carried out for targets much smaller than 
the wavelength; these yield, for a rigid target, 

T = 20 log , (56) 

where V is the volume occupied by the reflecting ob- 



362 


THEORY 


ject. Equation (56) is the so-called Rayleigh scatter¬ 
ing law. The echo intensity is directly proportional 
to the square of the volume of the target and in¬ 
versely proportional to the fourth power of the wave¬ 
length; thus, the echo intensity drops off rapidly as 
the wavelength increases. 

20.7 EFFECT OF PULSE LENGTH 

All the previous discussion in this chapter has been 
concerned with sound waves emitted in an essentially 
continuous fashion. While Section 2.3 discussed the 
effect of pulse length in terms of ray acoustics, this 
section will describe the effect of pulse length on the 
observed target strength in terms of wave acoustics, 
developed from the analysis in the preceding sections 
of this chapter. 

It was shown in Section 12.2 that at any instant 
the scattered sound energy received back at the 
transducer from the projected pulse comes from a 
spherical shell of thickness cr/2, where c is the sound 
velocity and r is the duration of the pulse. This result 
is still true on the more accurate wave theory pre¬ 
sented in Section 20.2, as long as the fluid is homo¬ 


geneous and the target is rigid and convex. From a 
concave target, sound reflected several times may 
arrive later than singly reflected sound. 

This thickness cr/2 is known as the pulse length. 
When the pulse length is so long that it includes many 
Fresnel zones, the echo level will be essentially the 
same as that observed for continuous sound, pro¬ 
vided the echo is measured at a time when the wave¬ 
lets are arriving from all these zones. At the beginning 
of the echo, when only the first few zones are con¬ 
tributing, and toward the end, when only the last 
zones return wavelets to the source, the echo struc¬ 
ture is more complicated. However, an application of 
the Fresnel zone theory would probably give correct 
results in this case. 

When the pulse is only a few Fresnel zones long, 
the echo structure is presumably more complicated, 
and the echo duration, for example, may be expected 
to exceed the duration of the outgoing pulse. The 
pulse length cannot be less than the thickness of a 
Fresnel zone, since in that case the outgoing pulse 
would consist of less than half a cycle, and the wave¬ 
length would cease to have much meaning. 



Chapter 21 


DIRECT MEASUREMENT TECHNIQUES 


S ubmarine target strengths have been calcu¬ 
lated theoretically and measured experimentally. 
The theoretical calculations described in Section 20.5 
are based on assumptions simplifying the geometry 
of the hull and conning tower, and the way in which 
the submarine reflects sound. Actual measurements 
in the field are necessary to verify and amplify these 
theoretical predictions and to assess their accuracy. 

Measurements have been both direct and indirect. 1 
Direct measurements consist of echo ranging, with 
short pulses of supersonic sound, on a submerged sub¬ 
marine at various ranges, depths, and speeds. The in¬ 
tensities of the received echoes are then measured and 
converted to target strengths. This chapter describes 
in detail the various experimental procedures and 
techniques employed by different laboratories in the 
direct measurements of submarine target strengths. 

Indirect measurements, on the other hand, use con¬ 
tinuous sound or light reflected from a scale model of 
a submarine, and interpret these results in terms of 
supersonic sound reflected from an actual submarine 
of the same shape; Chapter 22 describes how target 
strengths are measured indirectly. The results of both 
the direct and indirect submarine target strength 
measurements are presented and discussed in Chapter 
23 while both the techniques and results of target 
strength measurements on surface vessels are treated 
in Chapter 24. 

21.1 PRINCIPLES OF DIRECT 
MEASUREMENT 

In order to calculate target strengths, echoes from 
a submarine may be compared with echoes received 
at the same time and under the same conditions from 
a sphere. From the relative intensities of the echoes 
from the submarine and from the sphere, and from 
the expression for the target strength of a sphere 
[equation (10) in Chapter 19], the target strength 
of the submarine could be readily computed. Since 


only the relative intensities of two echoes would need 
to be determined, no absolute measurements or cali¬ 
brations would be required. But at sea, a sphere large 
enough to return a strong echo at ranges normally 
used in echo ranging is too awkward to handle easily 
and therefore cannot be used in practice to obtain 
target strengths. 

Instead, target strengths are always found bv using 
the fundamental definition [equation (6) in Chapter 
19], which defines the target strength of any object 
in terms of the echo level, the source level, and the 
two-way transmission loss from the projector to the 
target and back to the projector again, all expressed 
in decibels. This expression is simple and easy to use 
and has the advantage that all the quantities appear¬ 
ing in it may, in principle, be measured directly. Only 
the difference between the echo level and the source 
level, and the transmission loss which the signal un¬ 
dergoes as it travels from the projector to the tar¬ 
get need to be known in order to find the target 
strength. 

Unfortunately, the difficulties of calibration and 
other practical problems not yet resolved make the 
fundamental definition less useful than may be sup¬ 
posed. In particular, the calibration of the transducer, 
described in Section 21.4 as the measurement of its 
output as a projector and its sensitivity as a receiver, 
and the determination of the transmission loss, de¬ 
scribed in Section 21.5, as well as the large fluctua¬ 
tions and variations normally encountered in under¬ 
water sound experiments, introduce numerical un¬ 
certainties which cannot be accurately evaluated. 
Nevertheless, the fundamental definition of target 
strength introduced in Section 19.1.3 has been used 
in all direct measurements and has led to reasonably 
consistent results. 

21.2 EXPERIMENTAL PROCEDURES 

Four groups have measured submarine target 
strengths directly. They are: University of California 


364 


DIRECT MEASUREMENT TECHNIQUES 



-► ELECTRICAL CONNECTIONS 

-*_ ELECTROMECHANICAL CONNECTIONS 

Figure 1 . Experimental arrangement. 


Division of War Research at the U. S. Navy Elec¬ 
tronics Laboratory, formerly the U. S. Navy Radio 
and Sound Laboratory, San Diego, California 
[UCDWR3; Columbia University Division of War 
Research at the U. S. Navy Underwater Sound 
Laboratory, New London, Connecticut [CUDWR- 
NLL]]; Woods Hole Oceanographic Institution, 
Woods Hole, Massachusetts [WHOI]; and the 
Underwater Sound Laboratory, Harvard University, 
Cambridge, Massachusetts [HUSL], In addition, 
various groups at Fort Lauderdale, Florida, have 
also made measurements of this nature. Widely vary¬ 
ing procedures and techniques have been employed 
by these groups. 

21 . 2.1 San Diego 

Most of the direct measurements by UCDWR 
have been made off the coast of California aboard the 
USS Jasper (PYcl3), a converted 135-ft yacht built 
in 1938, which echo ranged on various S-boats or 
occasionally on new fleet-type submarines. Square- 
topped signals from 0.5 to 200 msec long were sent 
out, usually at a frequency of 24 kc and sometimes 
at 45 or 60 kc. Early trials used a QCH-3 magneto- 


strictive transceiver driven at a frequency of about 
24 kc; 2 a few measurements were also made with an 
experimental model of frequency-modulated sonar 
gear. 3 Later runs employed standard JK or QC trans¬ 
ducers 4 or specially designed equipment. 5 

Most of the echo-ranging equipment was installed 
in the wardroom of the Jasper; a schematic diagram 
of the installation is shown in Figure 1. A pelorus, an 
open sighting device attached to a dial and employed 
in determining bearings, was mounted topside on the 
flying bridge. An observer visually trained this pe¬ 
lorus on a float towed by the submarine, and the rela¬ 
tive bearing of the pelorus was relayed to a repeater 
dial in the wardroom below. Here, another observer 
followed the relative bearings of the pelorus and 
trained the transducer on them; obviously, the bear¬ 
ing accuracy obtainable in this manner was not very 
high. Then the echoes received by the transducer 
were amplified and fed into a cathode-ray oscilloscope 
to be photographed on continuously moving film by 
a high-speed camera. The echoes were also usually 
heterodyned and monitored over a loud speaker, and 
supplementary records were made on the sound- 
range recorder, where the keying interval was con¬ 
trolled manually as the range changed. 





























EXPERIMENTAL PROCEDURES 


365 


Two types of runs were made. In one, illustrated 
in Figure 2, the target strength was measured as a 
function of the aspect of the submarine; the sub¬ 
marine, usually at periscope depth, proceeded at 
creeping speed while the .Jasper circled it, trying to 
maintain a nearly constant range. The other, shown 
in Figure 3, comprised opening and closing runs, and 
was used to measure the echo level as a function of 



range to determine the transmission loss. Here, the 
submarine proceeded on a straight course while the 
Jasper followed a divergent course, bearing approxi¬ 
mately 60 degrees from the submarine and opened 
the range until contact was lost; then a closing run 
was made on a collision course down to a range of 
several hundred yards. During both opening and 
closing runs, the speed and course of the Jasper and 
the submarine were held so that the aspect which the 
submarine presented remained constant. 

Since these runs were made, a new type of fre¬ 
quency modulation sonar has been set up at the 
Sweetwater calibration station of TJCDWR for meas¬ 
uring the target strengths of small objects. 6 It is be¬ 
lieved that measurements may be made more quickly 
with this system than with the standard pinging 
system, but no results are available at the present 
time. 

21.2.2 New London 

At New London, tests were made by CUDWR 
aboard the USS Sardonyx (PYcl2) which echo- 
ranged in Long Island Sound on the USS S-48 
(SS159), a 1,000-ton S-boat 267 ft long, first com¬ 
missioned in 1922. 7 The submarine followed a straight 
course at a keel depth of 80 ft while the Sardonyx 
circled around it in an arc to maintain an approxi¬ 
mately constant range. 

A device was used automatically to range on 
center bearings. The amplified echo intensity was 
kept constant by manual control of the amplifier 
gain as the echoes were observed on a cathode-ray 
oscilloscope. Relative echo intensity was obtained by 


recording the amplifier gain settings and by referring 
to a calibration curve for the system. The bearing, 
course, and range of both vessels, and the gain 
settings were recorded about every half-minute. Be¬ 
cause complete calibration and transmission data 
were not available, absolute target strengths could 
not be computed. Instead, echo intensity was calcu¬ 
lated as a function of aspect in decibels relative to the 
echo level at an arbitrary aspect and plotted for 
ranges of 600, 1,000, and 1,200 yd. 

U?S JASPER 



\ 

\ 

\ 

\ 

\ 

\ 

Figure 3. Opening run. 

21.2.3 Woods Hole 

Target-strength measurements were also made by 
WHOI observers aboard the USS SC665 just off Fort 
Lauderdale, Florida. 8 ’ 9 Navy QCU sonar gear was 
employed, with pulses from 60 to 80 msec long sent 
out alternately at 12 and 24 kc, at slightly different 
signal lengths to facilitate separation of the 12-kc 
data from the 24-kc data. Apparatus was used to 
range on center bearing. 

A hydrophone nondirectional in the horizontal 
plane was mounted above the conning tower of the 
210-ft Italian submarine Vortice. Accessory recording 
equipment was installed aboard the submarine in 
order to measure the level of the received signals and 
to determine the transmission loss from the SC665 
to the submarine. The submarine proceeded on a 
straight course, while the SC665 circled the subma¬ 
rine to investigate aspect dependence, and opened 
and closed the range to investigate range dependence. 
The submarine also traveled at different speeds and 
different depths in order to ascertain possible varia¬ 
tion of target strength with the speed and depth of 
the submarine. 







36G 


DIRECT MEASUREMENT TECHNIQUES 


21.2.4 Harvard 

The target strength of the Italian submarine 
Vortice was also measured by HUSL workers using a 
special sonar first in the area of the Bahama Islands, 
then off the coast of Florida near Port Everglades. 
Sonar gear mounted aboard the USS Cythera (PY31) 
echo ranged on the submarine at a frequency of 
26 kc. 

The first series of tests was made near stern aspect 
as the Cythera and Vortice followed parallel courses 
at speeds from 2 to 6 knots. 10 Cut-ons were obtained 
by listening to the echoes. Very few data were col¬ 
lected; only 114 echoes were obtained on the Vortice 
during the two days of measurements so that the re¬ 
sults cannot be considered conclusive. 

During the second and more complete series of 
tests, the Cythera maneuvered around the Vortice in 
order to determine the dependence of target strength 
on aspect angle, altitude angle, and range. 11 The 
Vortice maintained a speed of 3 knots on a base course 
at depths of 100, 300, and 400 ft. Echo intensities 
were obtained for groups of approximately 10 echoes; 
the source level was measured by training the pro¬ 
jector at a monitor transducer, then feeding the 
voltage across the monitor transducer into a cathode- 
ray oscilloscope and finding the voltage that had to 
be applied to the oscilloscope in order to balance it. 
The speed of the Cythera was held close to that of 
the Vortice to prevent bearings from changing too 
rapidly; training the projector was accomplished by 
cut-ons. A vertically directional beam from a QHF 
transducer was used in addition to the original non- 
direetional beam. 

Aspect angles were estimated at intervals of 5 
degrees; ranges correct to about 25 yd were read from 
the sound-range recorder. Altitude angles were not 
recorded; instead, they were computed from the 
range, as read from the recorder, and from the depth 
of the submarine, measured from the ocean surface 
to the center of the control room about 12 ft above 
the keel of the submarine. 

21.2.5 Fort Lauderdale 

Three runs were made off the coast of Florida by 
observers from groups at Fort Lauderdale. In one 
series of tests, the YP451 remained stationary and 
echo-ranged on the USS Pintado (SS387) and the 
USS Pipefish (SS388), two new fleet-type submarines 
which ran past the YP451 at prearranged depths, 
speeds, and ranges. The equipment aboard the YP451 


included a crystal transducer, driven at 60 kc, which 
was suspended on a pendulous pipe so that it was 
15 ft below the surface. The platform carrying the 
transducer was stabilized by an automatic pilot gyro 
control in one dimension, with its horizontal axis of 
rotation normal to the axis of the sound beam. In 
addition, the transducer was automatically trained in 
elevation. The pendulum and gyro provided a plat¬ 
form which was stabilized in the most critical direc¬ 
tion, while the elevation control centered the sound 
beam on the target vertically; the transducer was 
trained manually on the target in the horizontal 
place. The beam width was roughly 25 degrees hori¬ 
zontally and 10 degrees vertically. A 6-string electro¬ 
magnetic oscillograph recorded the echoes. 

Unfortunately, operations with the YP451 were 
hampered by mechanical difficulties in the alternating 
current generator and by failure of radio communica¬ 
tion with the escort vessel which maintained sound 
communication with the submarine. Although this 
lack of communication resulted in unpredictable 
maneuvers by the submarine, fairly satisfactory data 
were obtained on echoes from the submarines. 

In the second and third runs, signals 30 msec long 
were sent out at a frequency of 60 kc every 0.6 sec. In 
the second run, a fleet-type submarine at periscope 
depth followed a straight course at a speed of 6 
knots. The echo-ranging transducer, mounted with 
accessory equipment in a submerged unit, circled 
about a fixed point, 230 yd from the course of 
the submarine, in a radius of 125 ft and at a depth 
of approximately 35 ft. Aspects were estimated 
trigonometrically from observed ranges, which had 
been corrected for the position of the echo-ranging 
unit in its turning circle. 

During the third run, an R-boat was the target, at 
a keel depth of 100 ft and a speed of 6 knots. The 
range was decreased continuously; cut-ons were em¬ 
ployed in training. Since the echo intensities varied, 
depending on where the beam struck the submarine, 
a series of echo maxima was obtained and was used 
to calculate the target strength. These maxima are 
illustrated in Figure 4, where the echo level — in 
decibels below the source level — is plotted against 
the range; each point represents an individual echo. 
The target strength was computed from the received- 
echo voltage, as measured on a film continuously ex¬ 
posed to a cathode-ray oscilloscope, hydrophone 
sensitivity, total power output into the water, 
directivity index of the transducer, and the esti¬ 
mated transmission loss. 



ANALYTICAL PROCEDURES 


367 



21.3 ANALYTICAL PROCEDURES 

Target strengths reported here were obtained for 
the most part from measurements of amplitudes of 
echoes recorded photographically. Sometimes, how¬ 
ever, operating conditions were so poor that upon 
examination of the photographs it was difficult to 
identify individual echoes and to distinguish them 
from noise signals. Therefore echo recognition is im¬ 
portant in the study of target strengths and is par¬ 
ticularly relevant to a discussion of analytical proce¬ 
dures. 

Echo recognition depends not on the intensity of 
the echo alone, but primarily on the difference be¬ 
tween the intensity of the echo and the intensity of 
the background. 12 At close ranges, echoes are usually 
strong enough to be easily recognized and are clear 
and well defined. Distant echoes, however, are often 
so weak that they cannot be distinguished from the 
background, and irregular spines and patches may 
effectively obscure the echo; hence a study of the 


structure of echoes from submarines and other tar¬ 
gets at short ranges may be useful in the recognition 
and identification of distant echoes from these same 
targets. Weak echoes may sometimes be attributable 
to poor training of the transducer or roll and pitch 
of the echo-ranging vessel, both of which may direct 
the sound beam away from the target. A high back¬ 
ground of reverberation and noise may make an echo 
hard to recognize. Rough seas and a wide transducer- 
beam pattern contribute to a high reverberation 
level, while a surface vessel at moderate speeds or a 
shallow submarine at high speeds may originate 
enough self-noise in the transducer dome to mask the 
echo. Reverberation is treated in detail in Chapters 
11 to 17. 

Once the echo is recognized and definitely identified 
as the desired echo, the problem becomes one of 
measurement and analysis. Various analytical pro¬ 
cedures have been employed by different groups in 
processing the raw material, from the oscillogram to 
the computed target strength. 






























































3G8 


DIRECT MEASUREMENT TECHNIQUES 


21.3.1 San Diego 

At San Diego, 35-mm film, running at a speed of 
either 2.5 or 12.5 in. per sec, was exposed to traces on 
a cathode-ray oscilloscope and then processed and 
read on an illuminated viewer. Peak echo amplitudes 
were measured in millimeters, corrected where neces¬ 
sary for the width of the spot of light on the oscillo¬ 
scope screen, and averaged over a series of echoes. The 
average was then converted to mean-square pressure 
level in terms of the calibration constants of the 
equipments. The transducer and accessory equip¬ 
ment were calibrated before and after each run with 
an auxiliary calibrated transducer, lowered on a boom 
from the side of the Jasper; Section 21.4 comprises a 
discussion of calibration errors. 

In computing target strengths at San Diego, cor¬ 
rection was also made for the deviation of the target 
from the axis of the sound beam on the basis of beam 
patterns measured in the laboratory. In addition, the 
range was found by measuring the distance between 
the midpoints of the echo and the signal on the film, 
and referring to index marks recorded every 50 msec 
at the bottom of the film, corresponding to range 
intervals of about 40 yd. From calibration data, the 
source level was calculated, which together with the 
echo level, and the transmission loss as measured dur¬ 
ing the opening and closing runs or, less accurately, 
estimated from prevailing oceanographic conditions 
but neglecting possible surface reflections, gave the 
target strength. Simultaneous sound-range recorder 
records provided a convenient check on the oscillo¬ 
grams. 

21.3.2 Fort Lauderdale 

A similar procedure was followed by the groups at 
Fort Lauderdale in analyzing the echoes obtained 
there. Here, however, the film moved more slowly, 
at a speed of approximately 1 in. per sec; only 50 ft of 
film could be accommodated inside the camera. Con¬ 
sequent^’, the echoes were compressed horizontally 
and were less detailed, but were still readily measur¬ 
able. 

The fine detail of the oscillograms made at San 
Diego enabled close determination of echo length as 
well as a study of echo structure for short pulses; this 
information was supplemented by an examination of 
echoes registered on the sound-range recorder. Target 
strength determinations from the films recorded at 
Fort Lauderdale, however, may be more accurate 


than that recorded at San Diego since the motion of 
the transducer was better controlled, the fluctuations 
smaller, and the values more consistent. 

21.4 CALIBRATION ERRORS 

Errors in target strengths measured directly must 
be due to errors in the echo level, the source level, or 
the transmission loss, since these target strengths are 
computed from equation (6) in Chapter 19. Incorrect 
echo level or source level determinations are usually 
attributed either to errors in calibration, or to errors 
in reading the echo level from the trace of the echo 
recorded oscillographicallv which UCDWR observers 
estimate as 2 or 3 db at the most. This section de¬ 
scribes errors attributable to calibration of the equip¬ 
ment; uncertainties in the evaluations of the trans¬ 
mission loss are discussed in Section 21.5. 


21.4.1 Purpose of Calibration 

In target-strength studies, the principal purpose of 
calibration is not so much the absolute determination 
of the source level and the absolute determination of 
the echo level, but rather the measurement of the 
difference between the two levels. In other words, it 
is necessary to know only the sum of the transducer 
output as a projector and response as a receiver if the 
echo level is measured in terms of the voltage across 
the terminals of the transducer. Then the difference 
between the echo level and the source level is simply 
the difference between (1) the echo level, in decibels 
above one volt, and (2) the sum of the projector out¬ 
put and receiver response of the transducer. 

The latter sum can be obtained by means of 
auxiliary transducers, without bothering about actual 
sound pressures. One scheme may employ an auxiliary 
hydrophone and an auxiliary projector. As a first 
step, the hydrophone could be lowered from a boom 
on the echo-ranging vessel, a few yards away from the 
transducer to be calibrated, and the transducer out¬ 
put measured in terms of the response of the hydro¬ 
phone. Then the auxiliary hydrophone and the trans¬ 
ducer, close together, are both exposed to sound from 
the auxiliary projector some distance away. Thus, 
the response of the transducer could be compared 
with the response of the auxiliary hydrophone; com¬ 
bining the measurements would give the desired cali¬ 
bration of the transducer. 



TRANSMISSION LOSS 


369 


21.4.2 Calibration Techniques at 

San Diego 

The methods of calibration most commonly used 
in target strength measurements, however, employ 
calibrated transducers; at San Diego, an auxiliary 
transducer is lowered over the side of the Jasper and 
used with the standard echo-ranging transducer. First 
one is used as the projector, then the other, and final 
calibration is accomplished by referring to the con¬ 
stants of the auxiliary transducer as calibrated at a 
separate measuring station. Unfortunately, this 
system is susceptible to errors at every step, so that 
too much reliance cannot be placed on the accuracy 
of the calibration. 

At San Diego, the greatest error in calibration is 
believed to be in the measurement of the output of 
the auxiliary transducer, which is used to calibrate 
the echo-ranging transducer before and after each 
run, as mentioned in Section 21.3.1. This auxiliary 
transducer is calibrated at intervals of roughly four 
months. Slow drifts of as much as 3 or 4 db have been 
detected for crystal transducers between calibration 
checks every three or four months; this drift may be 
responsible for part of the “variation” observed dur¬ 
ing target-strength runs, as described in Section 
21.6.1. However, since it was not practicable to con¬ 
trol or even measure all the factors entering into gear 
calibration, there is no direct evidence on which to 
base estimates of the overall calibration error of echo¬ 
ranging equipment. 

21.4.3 Observed Calibration Errors 

Recent indirect evidence suggests, however, that 
calibration errors as great as 12 db may occur. An 
example of such large calibration errors is evident in 
the results of San Diego echo-ranging tests on a 
sphere. 13-15 The sphere, 1 yd in diameter, was sus¬ 
pended 16 ft below the surface of the ocean at ranges 
from 24 to 166 yd; echoes from pulses from 0.5 to 
7 msec long were received on a JIv transducer. Target 
strengths computed from equation (6) in Chapter 19 
varied from —24 to +3 db, approximately 12 db 
above and below the theoretical value predicted from 
equation (10) in Chapter 19. Although the very low 
values are possibly the result of training errors, the 
very high values seem rather large to be attributed to 
errors in the estimated transmission loss, especially 
since the values as high as 3 db were found when the 
transmission loss was measured directly with a hydro¬ 
phone placed (1) close to the projector and then 


(2) close to the target. However, the possibility that 
the transmission loss at short ranges fluctuates by 12 
db cannot be ruled out at the present time. This large 
error must result either from large fluctuations in 
short-range transmission, or from errors inherent in 
the calibration of the gear, provided that the 
theoretical formula in Chapter 19 for the target 
strength of a sphere is applicable to direct measure¬ 
ments. 

To provide a check on the validity of this formula, 
an auxiliary hydrophone was placed a few yards from 
the sphere during this series of observations and was 
used to measure both the outgoing pulse and the re¬ 
turning echo. The mean target strength of 350 echoes 
was found by this method to be — 13.3 db, in unusu¬ 
ally close agreement with the theoretical value of 
— 12 db. A similar result was obtained at Woods 
Hole. Thus, the 12-db discrepancy observed when 
the JIv transducer alone was used is undoubtedly the 
result of errors in the estimated transmission loss, in 
calibration, or in both. That large systematic errors 
in these quantities may sometimes be present, even 
when careful checks are provided, is suggested by the 
anomalously high values found at San Diego for the 
target strength of a submarine at 60 kc, and the 
similar results obtained by Woods Hole at 12 and 
24 kc, both reported in Section 23.6.2. 

Large errors in calibration may result from 
(1) large-scale variability of the calibrated auxiliary 
units employed in methods involving absolute cali¬ 
bration at sea; or (2) gross deviations of the sound 
field from the theoretical inverse square law in cali¬ 
bration measurements at close ranges, because of 
interference with reflections from the hull or from 
other surfaces nearby. Neither of these explanations 
seems very likely. So far no really satisfactory ex¬ 
planation of the large internal inconsistencies in 
direct target strength measurements has been ad¬ 
vanced. Calibration of ship-mounted gear at sea re¬ 
mains one of the most troublesome of all underwater 
sound measurements. 

21.5 TRANSMISSION LOSS 

It has already been pointed out that much of the 
error in the direct measurements of target strength 
may be due to errors in the estimated transmission 
loss; probably a large part of the variability in ob¬ 
served target strengths arises from variability in the 
transmission loss. This quantity varies widely from 
hour to hour and from place to place and is seldom 
known accurately. 



370 


DIRECT MEASUREMENT TECHNIQUES 



RANGE IN YARDS 


Figure 5. Typical transmission anomaly at 24 kc for an isothermal layer 70 feet deep. 


The transmission loss II is defined as the loss in 
intensity, in decibels, as the sound travels between 
a point 1 yd on the axis of the sound beam from a 
small projector, and the target. If the medium 
through which the sound travels is ideal — if no 
sound is absorbed, scattered, or refracted, or re¬ 
flected from the ocean surface or bottom — then the 
intensity of the sound varies inversely as the square 
of the distance from the source, as pointed out in 
Section 19.1.1, and the transmission loss, in decibels, 
is simply 20 log r, where r is the range in yards. In 
this case the total transmission loss 2/7 as the sound 
travels to the target and back to the projector again 
is simply 40 log r. 

Thi s inverse square loss, however, is only a part of 
the total transmission loss of sound in water. Sound 
energy is absorbed by the water and dissipated as 
heat energy. Small particles in the water scatter the 
sound in all directions. Furthermore, as the beam is 
refracted by a temperature gradient, it is bent and 
the cross section of the beam changes in area, chang¬ 
ing the intensity of the sound correspondingly. 

To account for transmission loss due to absorption, 
scattering, and divergence arising from refraction, 
the transmission anomaly is defined as the difference 
between the total measured transmission loss, and 
the transmission loss due to divergence according 
to the inverse square law alone. In decibels, then, 

A = H — 20 log r, (1) 

where A is the transmission anomaly, H the total 


transmission loss, and r the range. A typical plot of 
the transmission anomaly against the range is il¬ 
lustrated in Figure 5. 

The transmission anomaly has been found to de¬ 
pend rather strongly on the prevailing oceanographic 
conditions and most particularly on the variation of 
the temperature of the water with depth. The water 
in the ocean is usually characterized by a mixed layer 
of nearly constant temperature down to a certain 
depth; below that depth, a decrease in temperature 
with depth, or thermocline, will appear. The trans¬ 
mission anomaly depends markedly on the depth to 
this thermocline as well as on the depth of the hydro¬ 
phone receiving the echoes. 

When the temperature difference in the top 30 ft 
of water is 0.1 F or less, the transmission anomaly 
may be considered a linear function of range (see 
Chapter 5). Hence, it is convenient to define an 
attenuation coefficient as the change in transmission 
anomaly with range. As a derivative, 


dA 



where a is the attenuation coefficient, A the transmis¬ 
sion anomaly, and r the range. Since in target- 
strength runs the attenuation coefficient is measured 
not as a derivative, but as an average over range in¬ 
tervals of 500 or 1,000 yd, a is usually taken as 

A 

a — — 


r 


(3) 















TRANSMISSION LOSS 


371 



0 500 1000 1500 2000 2500 

RANGE IN YARDS 

Figure 6. Target strength plot. 


In Figure 5, a amounts to about 4.5 db per kyd, at 
24 kc. It may be that actually 

A = ar + b (4) 

where b is a constant. Present data indicate, however, 
that b is probably negligible. 16 

By definition, the transmission anomaly includes 
the effects of reflection from the ocean surface. So 
little is known about surface reflection with any de¬ 
gree of certainty, however, that no attempt is made 
to include in equation (4) an additional term to take 
it into account. If surface reflection is appreciable, it 
may cause the constant b in equation (4) to be nega¬ 
tive, in effect decreasing the transmission anomaly 
and therefore the transmission loss itself, as described 
later in Section 21.5.4. 

21.5.1 Methods of Measurement 

Transmission loss may be measured in three ways. 
First, during an opening or closing run, the echo level 
in decibels above the source level may be corrected 
for geometrical divergence by adding 40 log r; then, 
the result may be plotted as a function of the range, 
as long as the submarine maintains a constant aspect. 
A typical plot of this nature is illustrated in Figure 6. 
Then the slope of the points represents twice the 
attenuation coefficient, and the intercept at zero 
range corresponds to the target strength. Such a de¬ 


termination presupposes that the target strength 
does not change over the ranges used. 

Second, before and after each run on a submarine, 
a transmission run may be made with an auxiliary 
surface vessel, in the usual manner, as described in 
Section 4.3.2. 

Third, the signals transmitted by the echo-ranging 
vessel may be received by a hydrophone mounted on 
the submarine, amplified, and measured. The trans¬ 
mission anomaly and attenuation coefficient may 
then be determined readily by correcting the level of 
the echo above the source for simple geometric 
divergence, and measuring the slope of the plot of 
the echo level against the range. 

But measuring the transmission loss in any one 
of these three ways is difficult. Aspects and speeds 
must be carefully maintained and measured, a pro¬ 
cedure particularly difficult for a submerged sub¬ 
marine. For reasons of safety, the echo-ranging vessel 
is advised not to approach the submarine closer than 
about 300 or 400 yd, and poor sound conditions often 
limit echo ranges to 1,000 yd or even less, especially 
off the coast near San Diego. When a transmission 
run is made with an auxiliary surface vessel, hori¬ 
zontal temperature gradients may result in a trans¬ 
mission loss between the projector and the hydro¬ 
phone suspended from the surface vessel which is 
different from that between the projector and the 
submarine. Another disadvantage of measuring the 



















372 


DIRECT MEASUREMENT TECHNIQUES 



0 10 20 30 40 50 60 70 80 90 100 

FREQUENCY IN KILOCYCLES 


Figure 7. Attenuation coefficient as a function of frequency. 


transmission loss by the latter method is the presence 
of four vessels in the operating area, that is, sub¬ 
marine, escort vessel, echo-ranging vessel, and trans¬ 
mission measuring vessel. The use of a hydrophone 
mounted on a submarine is a definite improvement 
but introduces new horizontal and vertical directivity 
problems as well as installation complications. 

21.5.2 Inadequacy of Transmission- 
Loss Measurements 

All three methods have been used to measure 
transmission loss during direct target strength tests. 
Where ample and consistent data have been taken 
by any one of these methods, the transmission loss 
calculated from these data has been used to evaluate 
the target strength. 

Often, however, data have not been consistent. 
During one run at San Diego, for example, the plot 
of the echo level, corrected for inverse square law 
spreading against range, indicated an attenuation 


coefficient of 19 db per kyd at a frequency of 60 kc 
while measurements aboard the submarine when 
analyzed showed a value of only 10 db per kyd. An¬ 
other identical run the following day gave values for 
the attenuation coefficient of 11.5 and 16 db per kyd, 
respectively, as measured by the two methods. Ap¬ 
parently the errors were not systematic. This lack of 
consistency between two methods was not infre¬ 
quent. Recent San Diego target strength measure¬ 
ments, however, based on transmission loss measured 
with a nondireetional hydrophone mounted on the 
submarine, have been more consistent; this method 
promises to eliminate much of the uncertainty in the 
evaluation of the transmission loss. However, the 
measurements reported 13-15 indicate that even this 
method does not eliminate systematic error in the 
determination of target strength, possibly because of 
peculiarities of transmission at short ranges, possibly 
because of calibration uncertainties. Certain 60-kc 
measurements on the USS S-37 (SS142) at San 
Diego gave a beam target strength of 28.7 db with a 


















TRANSMISSION LOSS 


373 


standard deviation of 8.5 db when an attenuation 
coefficient of 20 db per kyd was assumed; when the 
transmission loss measured aboard the submarine 
was used in the computations, the beam-target 
strength rose to 40 db with a much smaller standard 
deviation of 3.5 db. In most trials reported here, it 
was necessary to evaluate the transmission loss from 
an attenuation coefficient,estimated for each run from 
the echo-ranging frequency employed and sometimes 
from the prevailing oceanographic conditions. 

21.5.3 Estimating the Attenuation 
Coefficient 

The attenuation coefficient in sea water varies 
widely and depends primarily on the frequency of the 
echo-ranging sound and on the prevailing oceano¬ 
graphic conditions. 14 For example, in mixed water, or 
water of constant temperature, at least 50 ft deep, 
this coefficient is about 5 db per kyd at 24 kc, and in 
the neighborhood of 15 db per kyd at 60 kc. A plot of 
the attenuation coefficient against frequency for 
ideal sound conditions is reproduced in Figure 7 and 
represents a rough average of observations primarily 
at 20, 24, 40, and 60 kc; the increase in attenuation 
coefficient with frequency is quite marked and shows 
why it is impractical to use very high frequencies for 
echo ranging. 

The attenuation coefficient increases markedly 
with poor sound conditions. At 24 kc, it may be as 
high as 15 db per kyd under poor conditions, or even 
40 db per kyd under extremely bad conditions. 

Very few data are available at 60 kc on the varia¬ 
tion of the attenuation coefficient with oceanographic 
conditions. Empirical formulas have been derived for 
the attenuation coefficient at 24 kc, however, as a 
function of the depth of the thermocline. For a hydro¬ 
phone above the thermocline, 

170 

a = 3.5 + — , (5) 

and for a hydrophone below the thermocline 

260 

a = 4.5 + — , (6) 

where a is the attenuation coefficient in decibels per 
kiloyard and D is the depth in feet to the thermocline. 
The probable error is about 2 db per kyd. 16 As implied 
in Chapter 5, these empirical formulas are, in general, 
less suitable for predicting the attenuation coefficient 
than other methods based on a more quantitative 


classification of the variation of temperature with 
depth, because transmission anomaly-range graphs 
significantly depart from straight lines under certain 
conditions. However, equations (5) and (6) are suf¬ 
ficiently accurate for the present purposes. 

Early target strength measurements showed that 
different values of the transmission loss were obtained 
by different methods, as described in Section 21.5.2. 
Therefore, in most calculations representative values 
of 5 and 20 db per kyd at 24 and 60 kc respectively 
were taken for the attenuation coefficient. Much of 
the time no account was taken of the oceanographic 
conditions which prevailed at the time of the tests, 
however, with the result that the reported target 
strengths varied considerably. Examples of this varia¬ 
bility are given in Section 21.6 of this chapter. 

21.5.4 Surface Reflections 

Reflection of sound from the surface of the ocean 
is neglected in all calculations of target strengths. 
Such an effect would offset, in part, the loss in in¬ 
tensity caused by spreading and absorption. 

Perfect specular reflection from the surface would 
effectively double the intensity of the sound incident 
on the target and the intensity of the echo returned 
to the projector, under ideal conditions. In other 
words, it would reduce the transmission loss by 3 db 
each way, or by a total of 6 db from the projector to 
the target and back again. Thus, in equation (4) the 
constant b would equal —3 db at ranges of a few 
hundred yards or more. 

Some evidences of surface reflection have been 
found experimentally. At San Diego a number of 
oscillograms of echoes from submarines have shown 
peaks or “spines” at the beginning and end of each 
echo, separated by a relatively smooth echo of lower 
intensity; an example is shown in Figure 8 for an 
S-boat at beam aspect. 17 The first peak is attributed 
to direct reflection from the hull of the submarine 
alone when the first part of the pulse strikes the tar¬ 
get and is reflected back to the projector along the 
shortest possible path; the final peak comes from the 
ray reflected from the submarine to the surface and 
back to the projector, after the direct echo from the 
submarine has been received. In other words, the two 
spines are attributable to reflection along only one 
path, since there will be a short time at both the 
beginning and end of the echo when the sound travels 
only one path back to the transducer. The intensity 
of the intervening echo is consequently lower because 



374 


DIRECT MEASUREMENT TECHNIQUES 



Figure 8. Surface-reflected sound. 


of a combination of both constructive and destructive 
interference throughout the duration of the echo, be¬ 
tween direct and surface-reflected sound. 

A different effect produced by surface-reflected 
sound is also indicated by more recent information 
from San Diego. 18 During echo-ranging tests on a sub¬ 
marine from 90 to 200 ft deep, double echoes were 
observed, under certain conditions, on the chemical 
recorder and on the oscillograph — a strong primary 


echo followed by a faint secondary echo, illustrated in 
Figure 9. This appearance of double echoes suggests 
that some of the sound is reflected directly back to 
the projector to form a primary echo, while some of it 
is reflected vertically upward to the ocean surface, 
reflected by the surface back to the submarine and 
finally back to the projector to form a secondary 
echo. Quantitative data show that the lapse of time 
between the primary and secondary echoes is equal 
to the time necessary for the sound to travel up to 
the surface and back again, thus confirming this 
hypothesis. 

Although almost all the sound striking the surface 
is unquestionably reflected back into the water at 
some angle, the perfect specular reflection expected 
from a flat surface seems unlikely at sea. The nor¬ 
mally rough surface of the ocean and the presence of 
air bubbles tend to scatter the sound rather than 
allow perfect specular reflection at the surface. 
Further evidence minimizing the effects of surface 
reflection on target strength values is seen in the 
excellent agreement between the results of the direct 
measurements computed neglecting surface reflec¬ 
tions, and both the indirect measurements and 
theoretical calculations, where surface-reflected sound 
either does not appear or may be readily eliminated. 
Partly for this reason, surface-reflected sound is 
neglected in all target strength computations in 
Chapters 18 to 25. However, the results shown in 
Figure 2 of Chapter 9 and described in Section 9.2.1 
suggest that reflection from the ocean surface is 
frequently very nearly specular. More data are 
needed to clarify the exact importance of surface- 
reflected sound in practical echo ranging. At present, 
the resulting uncertainty of 6 db is about the same as 
the other uncertainties of observation in target 
strength measurements. 

21.6 VARIABILITY OF ECHOES 

Perhaps the largest source of uncertainty in target 
strength measurements arises from variability of 
echo intensity. Observed echoes vary widely in two 
ways (see Section 21.1). Gradual changes in echo in¬ 
tensity over a relatively long period of time from a 
few minutes to hours are called variations. Superim¬ 
posed on these variations are marked changes which 
occur from echo to echo and are called fluctuations. 
A large part of the variability of echo intensity is due 
to variability in the sound-transmitting character- 





VARIABILITY OF ECHOES 


375 



SECOND ECHO 
-PRINCIPAL ECHO 


50 MSEC 


S- TYPE SUBMARINE AT 100 FT DEPTH 



S" TYPE SUBMARINE AT 90 FT DEPTH 


Figure 9. Double echoes. 












376 


DIRECT MEASUREMENT TECHNIQUES 



Figure 10. Variations and fluctuations in sphere echoes. 


istics of the ocean (see Chapter 7). The remainder 
may be ascribed to such external causes as changes in 
the performance of equipment and changes in target 
aspect. 

21.6.1 ^ ariation 

Variations occurring over a sufficiently long time 
are very difficult to detect. Sometimes they result 
from gradual changes in the characteristics of the 
echo-ranging gear employed and may be detected 
each time the system is calibrated. 

More often, however, variation may be most promi¬ 
nent during a long run in the course of a single day 
or on successive days. At long ranges, changes in the 
transmission conditions in the water may be responsi¬ 
ble for some of the variation observed; horizontal 
temperature gradients may occur and cause changes 
in the value of the transmission anomaly. This effect 
may be most conspicuous at long ranges for two in¬ 
terrelated reasons. First, if the ranges are long the 
operating area is much larger, and horizontal differ¬ 
ences in temperature may be more likely. Second, 
since the transmission anomaly increases with 
range, variations attributable to slow changes in 
the transmission anomaly will be greatest at long 


ranges. At short ranges, much less is known about 
variation. 

Marked variation in the echo level was observed 
during the course of a number of runs during early 
echo-ranging tests on a sphere in San Diego. 2 The re¬ 
sults of one reel of film exposed to the sphere echoes, 
as shown on a cathode-ray oscilloscope, are repro¬ 
duced in Figure 10; pulses were sent out at intervals 
of 1.2 sec and the range of the sphere was about 
109 yd. Here, the ratio of the observed echo ampli¬ 
tudes to the echo amplitudes predicted from theory 
(in which transmission loss is taken into account) is 
plotted for each individual echo received. The short¬ 
term changes are most noticeable, but the slow up¬ 
ward slope of the average of the points is evidence of 
variations as defined here. The cause of this variation, 
however, is not known. 

Changes in the calibration of the equipment over 
a period of time, known as “drift,” are also responsi¬ 
ble for some of the variation observed. As pointed out 
in Section 21.4.2, slow drifts of 3 to 4 db have been 
observed between calibration checks at San Diego, 
at approximately four-month intervals, in a crystal 
projector. Just how much of the variation normally 
encountered can be attributed to drift, however, can¬ 
not be estimated very accurately. 





































































VARIABILITY OF ECHOES 


377 


21.6.2 Fluctuation 

Many factors contribute to the observed fluctua¬ 
tions of echoes. Much of t his rapid change in echo in¬ 
tensity may be ascribed to the roll and pitch of the 
echo-ranging vessel, which by changing the direction 
of the sound beam causes the received echoes to vary 
in intensity. Although gyroscopic stabilization of the 
transducer was employed at Fort Lauderdale to re¬ 
duce fluctuations arising from the roll and pitch of the 
ship, as described in Section 21.2.5, this system has 
not been used elsewhere for this purpose. Errors in 
training the echo-ranging transducer toward the sub¬ 
marine have also been responsible for some of the 
fluctuations encountered; training on the bearing of 
maximum intensity, by means of cut-ons, is approxi¬ 
mate and introduces variability in the received echo 
intensities by changing the direction of the beam 
relative to the submarine. 

In addition, surface reflection and interference 
phenomena may be expected to account for part of 
the fluctuations observed, as the sound beam fre¬ 
quently follows multiple paths to reach the submarine 
and return back again to the transducer. Chapter 7 
of Part I of this volume discusses the evidence show¬ 
ing that transmission fluctuations are very much re¬ 
duced when surface-reflected sound is minimized. 
Correlation has been observed between the depth of 
the transducer below the ocean surface, and the mag¬ 
nitude of the fluctuations observed in echoes from a 
sphere two ft in diameter; 2 at a range of the order of 
65 to 75 yd, elevating the transducer from a depth of 
50 to 10 ft below the surface increased the standard 
deviation of the echo intensity from 18 to 39 per cent. 
In addition, the overall fluctuation appears to de¬ 
crease as the signal length increases. At other than 
beam aspects, interference between echoes from dif¬ 
ferent parts of the submarine is undoubtedly respon¬ 
sible for part of the fluctuation observed, giving rise 
to an irregular “hashed” echo structure described in 
Sections 23.8.2 and 23.8.3. 

21.6.3 Effects oil Echo Level and 

Echo Structure 

Variation affects echo intensity; fluctuation affects 
both echo intensity and echo structure. Echo en¬ 
velopes never repeat exactly, and successive echoes 
at the same range, aspect, frequency, and signal 
length often appear totally different. This diversity 
of echo structure not only complicates measurement 


of the intensity of the echo, but also makes it difficult 
to resolve the length of the echo and the center of the 
echo, in effect preventing precise measurement of the 
range of an individual echo. 

Likewise successive echo intensities seldom repeat. 
As a result, some sort of average must be taken over 
successive echoes. If target strength is regarded as a 
measure of the fraction of the incident sound energy 
reflected by the target, the total reflected energy 
should be compared to the total transmitted energy. 
Such an analysis would require squaring the echo 
amplitudes to give the echo intensities, then inte¬ 
grating the intensities over the duration of the echo 
to give the total echo energy; this same procedure 
would be followed with the signal to yield the total 
signal energy. Such an analysis has not been found 
practical because of the complex instruments re¬ 
quired. In addition, it may be that aural and non- 
aural detection devices respond more to peak echo 
intensity rather than to total echo energy, and that 
therefore peak intensities are more significant. 

21.6.4 Peak versus Mean Eeho 
Intensity 

Since it has not been feasible to compare the re¬ 
flected and transmitted energy directly, peak echo 
amplitudes have been used to compute target 
strengths. Observations show that these average peak 
amplitudes do not differ significantly from the rms 
peak amplitudes, which would correspond to peak 
intensities. Thus the San Diego results may be re¬ 
garded as giving average peak intensities. Not only is 
this method simple and easy to apply, but also it pro¬ 
vides values which may be compared directly with 
recognition differential measurements where peak- 
echo intensities alone are considered. Peaks, however, 
fluctuate enormously, especially for off-beam echoes; 
a sample survey of 100 oscillograms of submarine 
echoes at San Diego showed a maximum fluctuation 
of 25 db between peaks, with fluctuations of 10 db 
not uncommon. 

An approximate comparison of reflected and trans¬ 
mitted energy might be made by measuring the mean 
echo intensity, averaged along the entire length of the 
echo, and correcting this intensity for the pulse length 
since the echo length generally is longer than the 
pulse length. Then the ratio of the signal and echo 
intensities, based on the same pulse length, would be 
equal to the ratio of the signal and echo energies. 

This procedure was attempted for six echoes 



378 


DIRECT MEASUREMENT TECHNIQUES 


recorded oscillographically at San Diego. 1 The echo 
amplitude was measured at small intervals along the 
length of the echo, squared to give the echo intensity, 
and averaged over the echo length as closely as the 
echo length could be estimated; then the enclosed 
area was calculated. Division of this area by the 
signal length gave a new intensity, the intensity 
which presumably would have resulted if the echo 
length had equaled the signal length. This sample 
analysis, although based on data not sufficient to 
warrant definitive conclusions, showed an insignifi¬ 
cant difference between peak echo intensities anti 
mean echo intensities corrected for signal length. 

In general, however, the peak echo intensity differs 
from the uncorrected mean echo intensity, and this 
difference is a function of the signal length. It was 
pointed out in Sections 19.3 and 20.7 that for long 
pulses, the echo will reproduce the signal envelope 
while for short pulses fluctuations in intensity will re¬ 
sult in an irregular structure, where sharp peaks 
stand out against a weak background. In the latter 
case, the peak echo amplitude may be considerably 
different from the mean echo amplitude, and may 
vary with signal length quite differently (see Sec¬ 
tion 23.5.1). 

The variability of echoes is responsible for a large 
part of the uncertainty in the echo level and trans¬ 
mission loss values which are used to compute target 


strengths. Since echoes are often so irregular that 
visual estimates of peak intensities are, at best, in¬ 
telligent guesses, UCDWR observers estimate that 
systematic errors of as much as 2 or 3 db may result 
from the difference in personal judgments of different 
observers. 

Because in practice fluctuations and variations be¬ 
have as very large accidental errors, only a statistical 
analysis of many echoes may be considered reliable. 
Hundreds of individual echoes must be carefully aver¬ 
aged, corrected, and analyzed to give target strength 
results of any significance. At San Diego, a camera 
has been installed aboard the Jasper to record, at the 
same time as each signal is transmitted, the roll and 
pitch of the vessel, the true bearing of the ship, the 
relative bearing of the transducer, and the time and 
pulse number. Such a record should be useful in 
analyzing and evaluating each echo, but so far has 
not been applied to a large number of measurements. 
So far target strength runs have been analyzed from 
a reasonably large number of individual observations; 
first, successive groups of five echoes each have been 
averaged, then an overall average computed con¬ 
sidering changes in transmission loss with range and 
changes in target aspect. Cumulative distributions 
and computations of probable errors and quartile 
deviations have been useful in interpreting the re¬ 
sults and assessing their reliability. 



Chapter 22 


INDIRECT MEASUREMENT TECHNIQUES 


R eflections from submarine models have been 
studied in order to discover the principal reflect¬ 
ing surfaces on a submarine and to measure sub¬ 
marine target strengths under controlled conditions. 
Both visible light and supersonic sound have been 
used in these model tests. 

In the investigation of reflection from submarines, 
models have many advantages over actual subma¬ 
rines. Generally, experimental conditions can be con¬ 
trolled much more easily under laboratory conditions 
than in the field. Laboratory use of carefully con¬ 
structed scale models makes possible a reasonably 
reliable evaluation of target strength as a function of 
aspect and altitude angles, as well as submarine class, 
and provides both a theoretical guide and a con¬ 
venient check on the direct measurements. 

22.1 PRINCIPLES OF INDIRECT 
MEASUREMENT 

Three groups have participated in the indirect 
measurements of reflections from submarine models; 
University of California Division of War Research at 
the U. S. Navy Radio and Sound Laboratory, San 
Diego, California [UCDWR]; Underwater Sound 
Laboratory, Massachusetts Institute of Technology, 
Cambridge, Massachusetts EMIT—USI/]; and the 
Underwater Sound Reference Laboratories, Columbia 
University Division of War Research, Mountain 
Lakes, New Jersey [USRL]. Only qualitative results 
were obtained at San Diego while actual target 
strength values were measured at MIT and at 
USRL. 

22 . 1.1 San Diego 

Early experiments were carried out at San Diego 1 
on a 1:60 scale model of the U570 or HMS/M 
Graph, a 517-ton German Type VIIC U-boat which 
was captured in 1941 off Iceland and served in the 
British fleet. The model, made of wood and finished 
with glossy white enamel, was illuminated by a 


standard projection bulb and photographed in vari¬ 
ous positions. 

The bulb was enclosed in a metal housing with a 
hole 134 in. in diameter on one side, and was placed 
as close as possible to the camera lens so that the 
angle between the incident and the reflected light at 
the submarine was only about 3 degrees. Photographs 
were made at different aspect angles, first with the 
submarine finished with enamel, then with the sub¬ 
marine covered in part by horizontal and vertical 
corrugations, and finally with coarse emery cloth 
covering certain areas on the model. The corruga¬ 
tions and emery cloth were affixed to the submarine 
in an effort both to reduce prominent reflections and 
to suggest locations for possible absorption treat¬ 
ment. 

These experiments were wholly qualitative, since 
no measurements were made. The main purpose was 
to discover the highlights on a submarine which 
might be largely responsible for strong reflections. 
Photographs for different aspect angles of the sub¬ 
marine model, without the corrugation or emery 
cloth, are reproduced in Figure 1. 

22 . 1.2 Massachusetts Institute of 
Technology 

Quantitative experiments using visible light re¬ 
flected from scale models were conducted at MIT- 
USL to calculate target strengths of four different 
submarines. 2 ' 4 A series of measurements were made 
on models of I1MS/M Graph, an old S-boat, the 
USS Perch (SS313) and the USS Sand Lance (SS381); 
these models were from 60 to 120 times smaller than 
the original submarines and were finished with a 
glossy black enamel. 

In order to compute the target strength of one of 
the submarines, light reflected from the submarine 
model was compared with light reflected from a 
sphere also enameled in glossy black. The target 
strength of the submarine was calculated from the 


379 



380 


INDIRECT MEASUREMENT TECHNIQUES 



Figure 1. Reflection of light from HMS/M Graph. 










PRINCIPLES Of INDIRECT MEASUREMENT 


381 


relative intensities of the light reflected from the sub¬ 
marine model and from the sphere, the scale factor of 
the submarine model, and the expression for the 
target strength of a sphere [equation (10) in Chap¬ 
ter 19]. 

The technique of these optical measurements was 
not simple. Light from a motion picture projector 
bulb passed through a polarizing element rotated by a 
synchronous motor and was focused on the submarine 
model. As a result, the plane of polarization of the 
incident light rotated at a high speed. Upon reflection 
from the model, the light passed through a second 
polarizing element and fell on a photoelectric cell; 
this second polarizing element was stationary, but 
adjustable. In effect, the two polarizing elements 
modulated the intensity of the light incident on the 
cell and made possible the use of a-c instead of d-c 
amplifying and measuring equipment. Moreover, the 
use of modulated polarized light greatly reduced the 
error caused by light scattered from the walls and 
other objects in the room in addition to the desired 
reflected light. 

At the same time, the photoelectric cell was also 
exposed to light from a neon lamp which was supplied 
with current from both a battery and a step-down 
transformer. As a result, the neon light contained a 
small a-c component whose intensity was directly 
proportional to the alternating current through the 
lamp, which was measured on a vacuum tube volt¬ 
meter. Since the light reflected from the model was 
adjustable in phase, by use of the second polarizing 
element, and since the light from the neon lamp was 
adjustable in magnitude, one was balanced against 
the other, thus canceling out the a-c component of the 
light reaching the photocell. When the a-c output 
from the photocell vanished, this condition of balance 
was obtained, and the voltmeter reading of the a-c 
lamp current was then proportional to the intensity 
of fhe model-reflected light. The use of this null 
method made it unnecessary to rely on a calibration 
of the photoelectric cell. 

To compute the target strengths, spheres from 1 to 
Y2 l /i in. in radius were substituted for the submarine 
models, and a similar procedure was followed. Photo¬ 
graphs were also made at different aspect angles and 
are illustrated in Figures 2 through 5. 

22.1.3 Mountain Lakes 

At Mountain Lakes, [New Jersey, a model of 
HMS/M Graph, similar to the model used at 


UCDWR and at MIT, was suspended in water in 
the path of sound from a supersonic transmitter. 3 The 
model, built to a 1:60 scale, was constructed of copper 
0.5 mm thick, plated with nickel 0.025 mm thick as a 
protection against corrosion. The model was sus¬ 
pended approximately 2)^ ft below the surface of the 
lake by wires at distances between 1 and 17 ft from 
the transducers, corresponding to full-scale target 
ranges between 20 and 310 yd. 

Pulses were not used in the indirect measurements 
at any of the laboratories. At Mountain Lakes, con¬ 
tinuous sound was transmitted by a quartz crystal 
projector, and the echo was received by a separate 
similar unit which served as a hydrophone. The model 
scale was 1:60. Since the importance of nonspecular 
reflection depends on the ratio of the wavelength to 
the dimensions of the target, it was necessary to 
scale the wavelength similarly. Consequently, an 
actual echo-ranging frequency of 24 kc, which is 
standard for most Navy gear, would require a 
frequency of 1,440 kc in tests with a 1:60 scale 
model. However, since the response of the trans¬ 
ducers was somewhat higher at higher frequencies, 
a frequency of 1,565 kc was used most of the time; 
the corresponding actual echo-ranging frequency 
was 26 kc. 

A beat-frequency oscillator, with a fixed frequency 
of 15 me, provided signals between 50 and 3,600 kc, 
which were amplified and sent through coaxial trans¬ 
mission lines to the projector. The received echo was 
amplified by a preamplifier in the hydrophone hous¬ 
ing, demodulated by the detector circuit and recorded 
on a continuous strip of paper as the submarine was 
slowly rotated about a vertical axis. The known cali¬ 
brations of the transducer and receiver were used, 
together with an assumed inverse square transmission 
loss to determine the target strength by using equa¬ 
tion (6) of Chapter 19. Under the controlled condi¬ 
tions possible at a reference station on a lake, the 
calibration is less difficult than it is for gear mounted 
on a ship at sea; thus the calibration error in these 
tests was probably small. Also, at such close ranges, 
temperature gradients and surface reflections are 
negligible. At a frequency of 1,565 kc, the attenua¬ 
tion coefficient predicted from Figure 7 in Chapter 21 
is about 0.6 db per yd. At ranges of only a few feet, 
this attenuation is negligible and the transmission 
loss may safely be assumed to obey the inverse 
square law. At ranges as great as 17 ft, however, this 
assumption may lead to target strengths which are 
about 6 db too low. 



382 


INDIRECT MEASUREMENT TECHNIQUES 



Figure 2. Reflection of light from HMS/M Graph. 

























PRINCIPLES OF INDIRECT MEASUREMENT 


383 



Figure 3. Reflection of light from S-type submarine. 







384 


INDIRECT MEASUREMENT TECHNIQUES 





fe- ^ V 




t ■ 



Figure 4. Reflection of light from USS Perch (SS313) 














PRINCIPLES OF INDIRECT MEASUREMENT 


385 



Figure 5. 


Reflection of light from USS Sand Lance (SS381). 





























386 


INDIRECT MEASUREMENT TECHNIQUES 


22.2 SUBMARINE REFLECTIVITY 

Since indirect target strength measurements are 
measurements not on actual submarines but on their 
scale models, certain possible corrections must be 
considered before the results can legitimately be 
compared with the results of the direct measure¬ 
ments. One possible source of error is in the reflec¬ 
tivity of the models used as compared with the re¬ 
flectivity of actual submarines. 

Since the experiments at UCDWR were qualitative 
in nature and designed only to determine the principal 
reflecting surfaces on a submarine, the question of 
absolute reflectivity is unimportant for those tests. 
The optical experiments at MIT, however, reported 
specific target strengths. These results were com¬ 
puted from the expression for the target strength of 
a perfectly reflecting sphere, in other words, a sphere 
which reflects all the sound striking it without trans¬ 
mission or absorption. Since both the submarine 
models and the spheres were finished in exactly the 
same way, these target strength results will apply 
only to perfectly reflecting submarines. 

At USRL, the reflectivity of the hull itself was 
found to be perfect, within experimental error, over 
the range of frequencies used. The hollow model was 
first tested filled with air, then filled with water. No 
difference was observed in the intensity of the re¬ 
flected sound for all frequencies between 50 and 2,000 
kc. Since reflection from an air-filled hull would be 
almost perfect, regardless of the transparency of the 
hull to sound, and since reflection from a water-filled 
hull submerged in water would come solely from the 
hull, with no air-water interface to reflect the sound, 
the experimental results did not justify assuming any¬ 
thing less than perfect reflectivity. Thus both the 
optical and acoustical indirect measurements are 
based on perfect reflection of the sound striking the 
submarine; transmission through the hull and ab¬ 
sorption in the steel are neglected. 

The steel hull of an actual submarine is also almost 
perfectly reflecting. Therefore, it appears that the re¬ 
sults of the indirect measurements may be inter¬ 
preted in terms of sound reflected from actual sub¬ 
marines. However, the presence of barnacles, moss, 
and other marine growth on the hull may appreciably 
affect the reflectivity. Such an effect would be im¬ 
portant for surface vessels or surfaced submarines, 
where the fouled hulls are exposed to the direct sound 
beam, but might not be significant for a submerged 
submarine, since the sound beam might not often 


strike the lower part of the hull where such growths 
attach themselves. No measurements have been 
made to ascertain the effect of barnacles and moss on 
the reflection of sound, but it is not believed to be 
significant. Therefore it appears that reflectivity 
considerations should not greatly affect any compari¬ 
son between direct and indirect measurements. 

22.3 WAVELENGTH EFFECTS 

If the indirect measurements of target strengths 
with submarine models are to be trusted, the experi¬ 
ments must be properly scaled, that is, the dimen¬ 
sions of the models, the ranges and depths at which 
the tests are made and all the wavelengths must be 
reduced by the same factor. This factor was 60 for 
the acoustical measurements at USRL, and all the 
quantities relevant to the measurements were changed 
by this factor. 

At MIT-USL, however, visible light was used. The 
models used in the optical experiments were from 60 
to 120 times smaller than the submarines they repre¬ 
sented. Assume an echo-ranging frequency of 24 kc, 
and the corresponding scaled wavelengths would be 
reduced to 0.1 cm for a 1:60 scale or 0.05 cm for a 
1:120 scale. Since the actual wavelengths employed 
were much shorter, errors might be expected in the 
results. 

Two errors in particular might be introduced. At 
certain aspects where the surface of the submarine 
subtends only a few Fresnel zones at 24 kc, as de¬ 
scribed in Sections 20.3 and 20.5, the model subtends 
many such zones, since the wavelength is much 
shorter compared with the dimensions of the sub¬ 
marine. As a result, the Fresnel integrals approach 
their asymptotic values, especially for surfaces of 
large radius of curvature, such as planes or cylinders, 
which subtend many Fresnel zones. Since the conning 
tower on a submarine is relatively flat, the optical 
measurements with very short wavelengths may 
overemphasize the effect of the conning tower. 

Secondly, nonspecular reflection is less than if the 
wavelength was properly scaled, by a factor equal to 
the square root of the ratio of the properly scaled 
wavelength to the improperly scaled wavelength 
actually used. This may account for the extremely 
low target strengths obtained optically at aspects 
giving very little specular reflection, such as the bow 
and stern. 

Diffuse reflection or scattering may be excessively 
large optically since the wavelength may be con- 



DIFFERENCES IN METHODS 


387 


siderably smaller than the surface irregularities. How¬ 
ever, this source of error has been minimized by the 
use of glossy black surfaces. 

22.4 DIFFERENCES IN METHODS 

Certain errors may arise from the differences in¬ 
herent between the direct and indirect techniques. 
As a submarine travels through the water, each sur¬ 
face may be assumed to be screened by a wake of 
some sort, or at least a turbulent condition in the 
water, and possibly also by air bubbles surrounding 
the hull and conning tower. 

Although this phenomenon may be present in the 
direct measurements of target strengths, it is absent 
in the indirect tests. In the optical methods, no re¬ 
flecting layer surrounded the submarine model; every 
effort was made to reduce reflection from dust par¬ 
ticles and from other objects in the room. In the 
acoustical tests, the submarine model was stationary 
throughout the measurements except for a very slow 
rotation in the horizontal plane, which could not give 
rise to wakes or air bubbles. The importance of this 
effect, of course, depends on the extent to which 
sound is reflected by turbulence in the water, which 
is negligible (see Section 34.3.2), or by air bubbles in 
the vicinity of the submarine (see Section 28.3.5). 

Extraneous reflections also may occur during the 
indirect measurements. Removal of the models, how¬ 
ever, has shown that the background level during 
both the optical and acoustical experiments is negligi¬ 
ble compared with the levels of the echoes from the 
models. 

Since continuous signals were employed during 
both the optical and acoustical measurements, sepa¬ 
rate transmitters and receivers had to be employed — 
a moving picture projector bulb and a photoelectric 
cell in the optical tests, and two similar transducers 


in the acoustical tests. The distance between them, 
however, was minimized, so that the angle of inci¬ 
dence and the angle of reflection at the model were 
as small as possible. At MIT, the bulb and photo¬ 
electric cell were approximately 14 in. apart, whereas 
the model was from 6 to 20 ft distant. At USRL, the 
two transducers were separated by less than 5 in., 
while the closest distance of the submarine model was 
11 in.; most of the measurements, however, were 
made with the source and receiver about 17 ft away 
from the model. 

Another difference between the direct and indirect 
measurements of target strength lies in the method of 
measurement. In the direct measurements, peak- 
echo amplitudes were used in all cases, since the 
echoes were short; in the indirect measurements, 
however, the echoes were continuous and the results 
were obtained by using rms intensities. The difference 
between mean intensities and peak intensities, and 
the dependence of this difference on pulse length are 
discussed in Chapter 21. 

Other errors may result from discrepancies in the 
construction of the models. Considerable difference 
was observed between the two models of HMS/M 
Graph, one used at MIT and the other at USRL, so 
that comparison of the two series of measurements is 
not completely justifiable. At some aspects, a differ¬ 
ence of G db in the target strengths of the two models 
was observed when optical measurements were later 
made on both models of HMS/M Graph ; these 
differences are described in Section 23.2.2. In addi¬ 
tion, rudders and propellers were missing from some 
of the models used at MIT and the model tested 
at USRL; at certain aspects, they may give rise to 
strong echoes. The models of the S-boat, the USS 
Perch and the USS Sand Lance, however, were sup¬ 
plied by the Bureau of Ships and are believed to be 
accurate. 



Chapter 23 


SUBMARINE TARGET STRENGTHS 


T arget strengths of submarines have been com¬ 
puted mathematically from the size and shape of 
a particular submarine, by the Fresnel zone method 
outlined in Chapter 20. They have also been meas¬ 
ured, both directly and indirectly, by use of the pro¬ 
cedures and techniques described in Chapters 21 and 



Figure 1 . Definition of angles. 


22, and have been studied in general as a function of 
orientation, submarine class, speed, range, pulse 
length, and frequency of the echo-ranging sound. 
This chapter presents the results of the different 
methods of determining target strengths of subma¬ 
rines and discusses their applicability to practical 
echo ranging. 

23.1 DEPENDENCE ON ORIENTATION 

Since a submarine is irregular in shape, the echoes 
which it returns depend markedly on its orientation 
with respect to the echo-ranging beam. The orienta¬ 


tion of such an irregular target is conveniently 
described in terms of aspect and altitude angles, 
defined in Figure 1. 

Consider a system of rectangular coordinates with 
the origin 0 at the center of the submarine. The aspect 
angle is defined as the angle between the x axis and 
the projection of the echo-ranging beam on the hori¬ 
zontal ( xz ) plane. It is measured in degrees from the 
bow of the submarine, in a clockwise direction as 
viewed from above; bow aspect is 0 degree, stern 
aspect 180 degrees, while beam aspect will be 90 and 
270 degrees for the starboard and port beams 
respectively. 

The angle between the echo-ranging beam and its 
projection on the horizontal (xz) plane is the altitude 
angle. It is measured in degrees, positive when the 
sound source is above the submarine, negative when 
it is below the submarine. If the projector is at the 
same depth as a level submarine, the altitude angle is 
0 degree; similarly, if it is directly above a level sub¬ 
marine, the altitude is 90 degrees. The vertex of both 
aspect and altitude angles is placed at the origin O 
of the coordinate system, which is taken at the 
geometric center of the submarine. 

23 . 1.1 Aspect Angle 

The strongest echo from a submarine is usually 
found within 20 degrees of beam aspect — between 
70 and 110 degrees, and between 250 and 290 de¬ 
grees, from the bow of the submarine. 1 These beam 
and near-beam echoes average about as strong as the 
echo from a sphere 35 yd in radius and correspond to 
a target strength of 25 db. Actually, target strengths 
as low as 7 db and as high as 40 db have been ob¬ 
served at beam aspect, directly and indirectly; most 
values, however, lie between 20 and 30 db. 

At other aspects, the target strength is much 
smaller and averages betw een 5 and 15 db, depending 
on the submarine and the altitude angle. At stern 
aspect, for example, target strengths measured di¬ 
rectly with standard gear vary from 4 to 19 db, de- 


388 








DEPENDENCE ON ORIENTATION 


389 


pending on the submarine 2-4 (see Section 23.2.1). 
Negative target strengths have been observed in the 
optical studies at certain aspects and altitudes; for 
example, at bow and stern aspects the target strength 
of the German U570 (HMS/M Graph ) varies from 

— 4 to — 6 db when the echo-ranging beam is below 
the submarine, at altitude angles between —5 and 

— 15 degrees. 5 Since such negative altitude angles are 
not encountered in practice when echo ranging from a 
surface vessel on a submerged submarine — because 


furthermore, the uncertainty in the aspect angle in 
some of the measurements was rather large. Conse¬ 
quently, the beam target strengths do not apply to an 
aspect angle of exactly 90 or 270 degrees. The altitude 
angle in all cases was small. In the direct measure¬ 
ments reported in this table, the submarine was sel¬ 
dom submerged to a keel depth greater than 100 ft, 
which at a range of 500 yd corresponds to an altitude 
angle of 4 degrees, while for the indirect measure¬ 
ments quoted the altitude angle was 0 degree. 


Table 1. Submarine target strengths. 





Beam 

Bow 

Stern 




target strengths 

target strengths 

target strengths 




and 

and 

and 



Frequency 

standard deviations 

standard deviations 

standard deviations 


Submarine 

in kc 

in decibels 

in decibels 

in decibels 

Theory 

U570 (HMS/M Graph ) 6 

25 

25.5 

11.5 

11.5 


Tambor class 7 


18.5 


8.5f 






l-5t 


USS S-28 (SS133) 8 

24 

18 




USS S-40 (SS145) 2 

24 

25 ± 4 

12.5 ± 4 

12.5 ± 4 


Fleet type 3 

24 

24 ± 5§ 

13+6 

19 ± 5 

Direct * 

Fleet type 3 

24 

29 + 3.5 



measurements 

USS S-34 (SS139) 9 

45 

25 




USS Tile fish (SS307) 9 

45 

26 




Fleet type 

60 

25 




R class 

60 



4 


R class 

60 



8.1 


British 4 

18 

29 




Vortice 10 

26 

42 

40 

40 


Vortice 11 

26 



10.5 

Indirect 

measurements 

S class 5 

USS Perch (SS313) 6 

USS Sand Lance (SS381) 6 


30 

30 

26 

6 

6 

5 

5 

5 

9 


U570 (HMS/M Graph ) 6 


27 

4 

i 


U570 (HMS/M Graph ) 42 

26 

25 

6 

6 


* San Diego measurements at 60 kc are not included here. § Average of all values in a 30-degree sector centered at beam aspect 

t Beam focused on conning tower. || Average of maximum values in the sector for each run. 

I Beam focused on screws. 


the projector is always above the target — these low 
target strengths are not very significant. Moreover, 
they may result from errors in the construction of the 
model (see Section 23.6.2) or from possible systematic 
errors inherent in the optical method (see Section 
22.4). 

Table 1 summarizes submarine target strengths at 
beam, bow, and stern aspects for the theoretical cal¬ 
culations 6 and for the direct 7-11 and indirect meas¬ 
urements. 2 ' 12 Certain controversial values discussed 
later in this chapter are omitted, as, for example, cer¬ 
tain San Diego measurements at 60 kc. Most values 
were averaged in sectors of roughly 10 or 20 degrees; 


Ranges varied from 200 to 1,000 yd, and the fre¬ 
quency from 12 to 60 kc for all tests except the optical 
studies at MIT, where the full-scale frequency was 
much higher. Although the results of the mathemati¬ 
cal studies are only approximate and the results of 
the direct measurements highly variable, all the data 
are generally consistent . 

The early San Diego values for a fleet-type sub¬ 
marine of the Tambor class 7 and an S-boat 8 are not 
reliable. An early experimental frequency-modulated 
gear was used to echo range on the Tambor -class sub¬ 
marine; since these results are difficult to interpret in 
terms of standard echo-ranging gear, they cannot be 






















390 


SUBMARINE TARGET STRENGTHS 


target strength in decibels 

bow 



STERN 

Figure 2. Aspect dependence (theoretical). 


weighted as heavily as other measurements. An ap¬ 
preciable variation of target strength with aspect 
angle was observed, in agreement with other meas¬ 
urements. Also, an observed difference in these ex¬ 
periments of bet ween 10 and 16 db in target strengths 
at beam and stern aspects, depending on where the 
beam was focused, seems confirmed by more reliable 
results. The actual values, however, must be con¬ 
sidered doubtful. 

The target strengths of the S-class submarine taken 
from reference 8 were measured by comparing sub¬ 
marine echoes with the echoes from a submerged 
sphere at a much shorter range than the submarine; 
fluctuations were large (see Figure 10 in Chapter 21), 


and the transmission loss was not known accurately. 
Furthermore, no significant variation in target 
strength with aspect angle was observed for the 
S-boat; this result alone makes the reliability of these 
measurements very dubious. 

The remaining values in Table 1 are in moderately 
good agreement with each other, especially at beam 
aspect; values for which no reference is given were 
found at Fort Lauderdale. The only results out of 
line are those from the Woods Hole measurements on 
the Italian submarine Yortice . 10 

These measurements on the submarine Yortice were 
made at a range of 1,000 yd and frequencies of 12 and 
24 kc; the submarine was proceeding at 6 knots at a 





























DEPENDENCE ON ORIENTATION 


391 



Figure 3. Aspect dependence (San Diego). 


depth of 150 ft. The values reported are of the order 
of 40 db, so much larger than any reported previously 
by any method that they appear to be the result of 
faulty calibration; the systematic error of about 
20 db at all aspects is difficult to explain in any other 
way. 

It might be pointed out, however, that the trans¬ 
mission loss as measured aboard the submarine was 
about 15 db greater than that expected from the pre¬ 
vailing oceanographic conditions. Therefore an at¬ 
tenuation coefficient of 4 db per kyd was assumed, in 
addition to inverse square divergence. In spite of 
such a transmission loss, however, the target strengths 


are more than 10 db greater than the highest values 
previously observed on a similar submarine. 

Figures 2 to 6 show typical variations of target 
strengths with aspect angle. Figure 2 is a plot of the 
theoretical calculations of the target strength of the 
U570 {Graph) for an echo-ranging frequency of 25 kc 
and a range of 1,000 yd. 6 They are based on approxi¬ 
mating the submarine by an ellipsoid of appropriate 
dimensions, with a conning tower of “tear drop” 
cross section; details of this method are described in 
Section 20.5. 

Figure 3 shows the result of a typical target 
strength run at 24 kc on a fleet-type submarine at 































392 


SUBMARINE TARGET STRENGTHS 



Figure 4. Aspect dependence (New London). Curves (range in yards): (A) 600; (B) 800; (C) 1,000; (D) 1,200. 


San Diego. The submarine followed a straight course 
at about 2Y 2 knots at periscope depth, while the 
Jasper circled it at a range of about 500 yd (see 
Figure 2 in Chapter 21). Each point on the curve is 
the average of all echoes obtained within the 15-de¬ 
gree sector centered at the indicated aspect angle, and 
represents, on the average, about 40 individual 
echoes. 

Figure 4 is a plot of the echo level against aspect 
angle for a series of tests made on the USS S-48 
(SS159) at New London and described in Section 
21.2.2. 13 Each contour represents measurements made 
at a different range. Since no correction was made 
either for the transmission loss or for the calibration 


of the equipment, no absolute target strengths are 
plotted. These echo level values cannot be compared 
directly with other target strength values, since 
they are relative to an arbitrary level. However, 
the differences in echo levels at different aspects 
correspond to the differences in target strengths at 
different aspects, as long as the range remains 
constant. 

Indirect measurements of target strength are il¬ 
lustrated in Figures 5 and 6. Figure 5 shows target 
strength as a function of aspect angle for a model of 
the USS Perch (SS313), as measured optically at 
MIT. 2 The altitude angle was 0 degrees and the full- 
scale range 600 yd. The results of the acoustical tests 































DEPENDENCE ON ORIENTATION 


393 


TARGET STRENGTH IN DECIBELS 
BOW 



STERN 

Figure 5. Aspect dependence (MIT). 


made at Mountain Lakes on a model of the U570 are 
reproduced in Figure 6, for an altitude angle of 0 de¬ 
gree, a full-scale range of 340 yd, and a full-scale 
frequency of 2G kc. 12 Bow and stern target strengths 
in Figures 5 and 6 are considerably lower than in 
Figure 2, probably because the ellipsoid used in the 
theoretical calculations was rounded at either end 
while the optical and acoustical models were pointed. 
Other discrepancies at bow and stern are discussed 
in Sections 23.8.2 and 23.8.3. 

23.1.2 Altitude Angle 

Target strength varies with altitude angle, but in 
most cases this variation does not appear to be im¬ 


portant practically. Figure 7 illustrates the theoretical 
predictions of the target strength of the U570 at beam 
aspect as a function of altitude angle for a range of 
about 1G yd; 5 since target strengths were found only 
for certain intervals of the altitude angle, they are 
represented as sectors in the polar plot of Figure 7. 
The sharp sectors at particular altitudes are attrib¬ 
uted to the sum of two separate reflections — from 
the blister tank and from the hull itself —■ neglecting 
interference phenomena. Figure 8 shows the same 
plot for the optical measurements made on a model 
of the USS Perch (SS313) for echo-ranging distances; 
the peak at 90 degrees, when the projector is directly 
above the submarine, arises from a strong reflection 





























394 


SUBMARINE TARGET STRENGTHS 


TARGET STRENGTH IN DECIBELS 
BOW 



STERN 

Figure 6. Aspect dependence (Mountain Lakes). 


from the deck of the submarine. The absolute values 
in Figures 7 and 8, however, are not comparable be¬ 
cause the theoretical calculations were carried out for 
a projector very close to the submarine, while the 
optical measurements applied to the ranges of several 
hundred yards usually encountered in practical echo 
ranging. 

Figure 9 is a smoothed curve showing the relative 
target strength of the Italian submarine Vortice 
plotted against aspect angle for altitude angles of 
0 to 10, 10 to 20, 20 to 45, and 45 to 90 degrees, as 
measured by Harvard observers; 14 for each curve, 
the relative target strength at beam aspect was 
arbitrarily set at 25 db. These data were obtained at 


a frequency of 26 kc, for submarine depths of 100 to 
400 ft and ranges up to 1,000 yd. The aspect de¬ 
pendence apparently becomes less marked and the 
curve smoother as the altitude angle increases. It 
might be pointed out, however, that for altitude 
angles greater than 20 degrees, and a submarine 
depth of about 400 ft, the sound beam does not com¬ 
pletely cover the submarine at near-beam aspects. 
Therefore the target strength would be expected to 
show less aspect dependence. 

Figures 10 and 11 show target strength aspect 
curves for different altitude angles, as measured in¬ 
directly. Optical measurements on a submarine of the 
S class are given in Figure 10 for altitude angles of 0, 
































BEAM TARGET STRENGTH IN DECIBELS 


DEPENDENCE ON ORIENTATION 


395 


ALTITUDE 
ANGLE IN 
DEGREES 



ALTITUDE 
ANGLE IN 
DEGREES 



Figure 7. Altitude dependence (theoretical), 16-yd 
range. U570 (HMS/M Graph). 


Figure 8. Altitude dependence (optical), 193-yd 
range. USS Perch (SS313). 



0 30 60 90 120 150 180 

BOW BEAM STERN 

ASPECT ANGLE IN DEGREES 

Figure 9. Target strength-aspect curves at different altitudes, for the Italian submarine Vortice (direct measurements). 














































390 


SUBMARINE TARGET STRENGTHS 



O 30 60 90 120 150 180 

BOW BEAM STERN 

ASPECT ANGLE IN DEGREES 


Figure 10. Target strength-aspect curves at different altitudes (optical). 


15, and 45 degrees. Figure 11 is a similar plot for the 
acoustical measurements on the model of the U570 at 
much smaller altitude angles of 0, 1.0, and 1.8 de¬ 
grees. Occasionally a pronounced maximum of the 
target strength has been observed at a very small and 
critical altitude angle, where the conning tower gives 
a prominent specular reflection or highlight. An ex¬ 
ample may be seen in Figure 11, at an aspect angle 
of about 105 degrees. Another example, at only one 
aspect angle, is shown in Figure 12, where the target 
strength of the U570 at bow aspect is plotted for five 
different altitude angles used at Mountain Lakes; 
here a strong reflection from the conning tower at an 
altitude of 1.0 degree results in a target strength of 
19 db, while the target strengths at altitude angles 
only a fraction of a degree different are considerably 
lower. Such maxima, however, are not common, and 
are generally confined to such a small sector of alti¬ 
tude angles that most of the time they are not likely 
to be observed in actual echo ranging. Their possible 


effect on the direct measurements has not been veri¬ 
fied because of the wide fluctuations in echo level 
tending to obscure such fine detail. 

Negative altitude angles are not encountered for 
surface vessels echo ranging on submarine targets. 
The low target strengths at these negative angles, 
measured in the optical studies, have already been 
mentioned in the preceding section. 

At very large positive altitude angles, the differ¬ 
ences between beam, bow, and stern target strengths 
are less marked and the resulting target strength- 
aspect curve is considerably smoother, as illustrated 
in Figure 9 for altitude angles between 0 and 90 de¬ 
grees, and in Figure 10 for altitude angles of 15 and 
45 degrees. Direct measurements on a deep subma¬ 
rine were also made at San Diego on the USS Tilefish 
(SS307) at a depth of 400 ft, 9 and gave results not 
significantly different from measurements at shal¬ 
lower depths except at quarter aspects. Values be¬ 
tween 25 and 27 db were obtained at beam aspects in 

























DEPENDENCE ON CLASS 


397 



BOW BEAM STERN 

ASPECT ANGLE IN DEGREES 

Figure 11. Target strength-aspect curves at different altitudes (acoustical). 


good agreement with values at periscope depth. At 
an aspect angle of 150 degrees, however, a target 
strength of 27 db was obtained, much higher than 
any other values reported at that aspect, directly or 
indirectly. -An S-boat, for example, measured in the 
same series of runs, gave a target strength of 14 db 
at the same aspect. 

This high value of 27 db may result from overcor¬ 
recting for the transmission loss by assuming an 
excessively large attenuation coefficient, since the 
range of the Tilefish was 760 yd compared to 150 yd 
for the corresponding measurements on the S-boat. 
The attenuation coefficient assumed was 10 db per 
kyd of sound travel at a frequency of 45 kc. However, 
high target strengths may actually be characteristic 
of deep submarines at certain aspects. It may be men¬ 


tioned that at Fort Lauderdale, one of the strongest 
series of echoes was obtained when echo ranging at 
stern aspect on a fleet-type submarine submerged to 
a depth of 250 ft, at ranges between 220 and 700 yd. 

23.2 DEPENDENCE ON CLASS 

Submarines of different sizes and shapes may be 
expected to reflect sound differently; the echoes from 
an R-boat 186 ft long and those from a new fleet-type 
submarine more than 300 ft long are not likely to be 
the same. In particular, specular reflection of sound 
from a submarine would be expected to depend 
rather critically on the shape of the hull and espe¬ 
cially on the different radii of curvature. 


























398 


SUBMARINE TARGET STRENGTHS 


12 db; and for a large fleet-type submarine about 19 
db. At beam aspect, the target strengths are more 
nearly the same. 

Unfortunately, no comparative measurements have 
been made on two different vessels by a single 
group during a single operation. In view of the sys¬ 
tematic errors of 5 to 10 db that may be present in 
target strength determinations, suggested in Sections 
21.4, 21.5, and 21.6, the differences shown in Table 2 
are not too conclusive. More accurate data are re¬ 
quired to allow any conclusion to be drawn about the 
variation of target strength between different sub¬ 
marines. 

23.2.2 Indirect Measurements 

Table 3 lists target strengths for different sub¬ 
marines measured indirectly at an altitude angle of 
0 degrees. These values were obtained under con¬ 
trolled conditions and are more self-consistent than 
the values measured directly; as a result, fluctuations 
are smaller and the differences between the values are 
probably more significant than in the direct measure¬ 
ments. Beam target strengths may vary between 25 
and 30 db; at stern aspect, the limits are 1 and 9 db. 


Table 2. Dependence of target strength on submarine class (direct measurements). 


Submarine 

Reference 

Frequency 
in kc 

Length 
in feet 

Beam 

target strength 
and 

standard deviation 
in decibels 

Bow 

target strength 
and 

standard deviation 
in decibels 

Stern 

target strength 
and 

standard deviation 
in decibels 

R class 


60 

186 



4 

R class 


60 

186 



8.1 

S class 

2 

24 

219 


12.5 + 4 

12.5 + 4 

S class 

Average from 
Table 4 

24 

219 

19.7 ± 2.5 



Fleet type 

3 

24 

304 

24+5 

13 ± 6 

19 + 5 

British 

4 

18 

300 

29 





Figure 12. Altitude dependence (Mountain Lakes). 


23.2.1 Direct Measurements 

Representative target strengths of different sub¬ 
marines measured directly are listed in Table 2. 
S-boats, R-boats, and fleet-type submarines have all 
been targets in direct measurements, but the experi¬ 
mental error is so great that it generally obscures any 
possible correlation of target strength with submarine 
class. However, Table 2 shows an apparent depend¬ 
ence of target strength on submarine class at stern 
aspect. The target strength of R-boats at stern aspect 
varies from 4 to 8 db; for an S-boat it averages about 


The possibility that a fleet-type submarine may re¬ 
turn a strong echo at stern aspect, which is suggested 
in Table 2, is confirmed by the optical measurements 
on the USS Sand Lance (SS381) quoted in Table 3. 
These measurements give a stern target strength of 
9 db, which is less than most similar values measured 
directly but still considerably larger than for any 
other submarine tested optically at that aspect. 
Since the indirect measurements at off-beam aspects 
are much lower than the direct measurements at 
those aspects, however, these optical values are 
probably not too significant. 























DEPENDENCE ON CLASS 


399 


Table 3. Dependence of target strength of submarine class (indirect measurements). 


Submarine 

Length 
in feet 

Beam 

target strength 
in decibels 

Bow 

target strength 
in decibels 

Stern 

target strength 
in decibels 

S class 6 

227 

30 

6 

5 

USS Perch (SS313) 5 

308 

30 

6 

5 

USS Sand Lance (SS381) 5 

310 

26 

5 

9 

U570 (HMS/M Graph)* 

220 

27 

4 

1 

U570 (HMS M Graph ) 12 

220 

25 

6 

6 



o 30 GO 90 120 150 180 

BOW BEAM STERN 


ASPECT ANGLE IN DEGREES 

Figure 13. Comparison of theoretical, optical, and acoustical target strength of the U570 (HMS/M Graph). 


Each submarine class will often evidence its own 
peculiar reflecting characteristics at certain aspects 
and altitudes. A striking example of this is the value 
of 20.2 db for the target strength of the U570 at an 
aspect of 120 degrees, reported in the Mountain 
Lakes measurements. Contrast this value with a 
maximum target strength of 20.3 db for beam as¬ 
pects, 9.4 db astern, and 17.4 db bow-on. This result 
is attributed to a strong specular echo from the back- 
plate of the conning tower, but is not confirmed by 


optical tests, perhaps because of differences in the two 
models. 

To test the reliability of these model results for 
each class of vessel, a comparison may be made be¬ 
tween results obtained by different indirect methods 
on the same type of submarine. Such a comparison is 
made in Figure 13, which shows target strength 
plotted against angle for the U570, as calculated 
theoretically and measured optically and acous¬ 
tically. 



































400 


SUBMARINE TARGET STRENGTHS 



0 30 60 90 120 150 180 

BOW BEAM STERN 

ASPECT ANGLE IN DEGREES 

Figure 14. Optical comparison of two models of U570 (HMS/M Graph). 


The theoretical results at off-beam aspects are 
not very realistic, since they assume a perfectly 
smooth and curved reflecting surface. In Figure 13, 
port and starboard sides were averaged in the optical 
and acoustical measurements to simplify the illus¬ 
tration. A considerable difference is evident between 
these three curves, which may be attributed in part 
to the errors in the models as well as to possible dif¬ 
ferences arising from the different methods used. 

Since different models of the U570 were used at 
MIT and at Mountain Lakes, the target strength 
of each model was measured optically, after the 
model used at Mountain Lakes had been finished in 
the same way as the MIT model. The results are 
reproduced in Figure 14 for the model originally used 
at Mountain Lakes and for the model first tested at 
MIT. Unfortunately, the Mountain Lakes model 
was not measured at aspects within 30 degrees of the 
beam, so that the comparison is not complete. Since 
the beam target strengths measured both optically 
and acoustically agree quite well, however, the tar¬ 
get strengths of both models are probably the same 
at beam aspect. 

The differences between the two curves in Figure 


14 cannot be ascribed to differences in the methods, 
since both models were tested in the same way —- 
optically. Instead, these differences, which near the 
bow are as great as 5 db, must be due to differences 
in the models themselves. Generally higher values 
were obtained for the Mountain Lakes model. In 
particular, diving planes at the bow, and horizontal 
and vertical rudders at the stern increased bow and 
stern reflections from the Mountain Lakes model. 
The MIr model had a knife-edge finish at bow and 
stern. Furthermore, the difference in the profile of 
the conning towers and the supporting structures for 
thetwomodelswould be expected to reduce somewhat 
the reflections of the MIT model at stern aspect. 

In view of these discrepancies between different 
models of the same submarine, and between the 
different indirect methods of determining target 
strength, too much reliance cannot be placed on the 
general differences in submarine classes shown in 
Tables 2 and 3. 

23.2.3 Asymmetry 

To a first approximation, the shape of a submarine 
may be represented by an ellipsoid which is sym- 






























DEPENDENCE ON CLASS 


401 



Figure 15. Target strength asymmetry. Target strength in decibels. 


metrical with respect to all three planes in Figure 1. 
This approximation was the basis of the theoretical 
predictions described in Section 20.5. If now the 
conning tower is added to the submarine immedi¬ 
ately above its center, the submarine loses its sym¬ 
metry about the horizontal ( xz ) plane. This asym¬ 
metry about the xz plane is largely responsible for 
the asymmetry in target strength-altitude angle plots 
for positive and negative altitude angles. In addi¬ 
tion, the shape of fuel and ballast tanks are fre¬ 
quently not symmetrical about the horizontal plane. 

Since the shape of the bow of a submarine generally 
differs from the shape of its stern, perfect symmetry 
about the yz plane is also absent. However, the port 


and starboard side of a submarine are, usually, nearly 
identical, with only very minor differences, to a first 
approximation, and therefore symmetry with respect 
to the xy plane may be expected. In other words, 
plots of target strengths as a function of aspect 
should be symmetrical about the longitudinal axis of 
the submarine. 

In the theoretical calculations of the target 
strength of the U570 as a function of aspect angle, 
symmetry about the xy plane was assumed, since the 
submarine was approximated by an ellipsoid of 
revolution. 

A lack of symmetry about the xy plane has been 
observed in both direct and indirect target strength 

























402 


SUBMARINE TARGET STRENGTHS 


measurements, and is apparent in Figures 3 to 6. 
Much of it is due to experimental error, and may not 
be real. Repeatable asymmetry in target strength- 
aspect curves has occasionally been found at San 
Diego and is illustrated in Figure 15 for three runs on 
an S-boat, where a sharp dip is evident just off the 
port bow at an aspect angle of about 340 degrees. 
This dip may be attributable to particular features 
of the construction of the S-boat, such as the lack of 
any surfaces normal to the sound beam so that specu¬ 
lar reflection cannot occur at that aspect. It is also 
possible that this decrease in target strength is a 
characteristic of bow echoes and that the aspect 
angles were in error by as much as 20 degrees; low 
bow target strengths are conspicuous in the re¬ 
sults of the optical measurements. On the other hand, 
the variability of echo intensities at other aspects is 
so large that this dip may not be real, even though it 
appeared during all three runs. 

Horizontal asymmetry in the indirect measure¬ 
ments is apparent in Figures 5 and 6. This asym¬ 
metry may be attributed largely to the asymmetrical 
models used, in other words, the port and starboard 
sides were not the same. Not only were the models 
asymmetrical, but, as shown in Section 23.2.2, models 
of the same submarine appeared to differ rather 
markedly. Because of these model differences, the 
observed data cannot be used to confirm the exist¬ 
ence of asymmetry in the target strength in the hori¬ 
zontal plane. 

23.3 DEPENDENCE ON SPEED 

Almost no data are available on the variation in 
target strength with the speed of the submarine. 
If echoes come only from the hull and conning tower 
of the submarine, it can be argued theoretically that 
the target strength of the submarine itself should not 
change as the speed is changed. But if a layer of air 
bubbles immediately surrounding the submarine con¬ 
tributes appreciably to the echoes received, then the 
target strength would be expected to depend on the 
speed and depth of the submarine; the scattering of 
sound from bubbles is discussed in Chapters 26 to 
35. In addition, turbulence in the water adjacent to 
the submarine, which would depend on the speed of 
the submarine, may be responsible for part of the 
reflection of sound. 

So far, little evidence has been uncovered to iden¬ 
tify a layer of air or turbulent water surrounding the 
submarine as an effective reflector of sound (see Sec¬ 


tion 33.3), although bubbles have been observed on a 
submerged submarine traveling through the water 12 
(see Section 27.1.1). Most direct measurements have 
been made on submarines at a creeping speed be¬ 
tween 1 and 3 knots; none have been made on a 
stationary, balanced submarine. Some data describe 
tests at 6 knots, but no significant difference has been 
observed between these results and results at lower 
speeds. At 6 knots, however, reasonably strong 
echoes were received from a wake behind a fleet- 
type submarine at periscope depth. As the sound 
beam crossed the submarine from bow to stern, 
echoes grew stronger, faded and died out completely 
for a short time. Then strong echoes were received 
for several hundred yards behind the submarine, 
which were attributed to reflection from the wake. 

23.4 DEPENDENCE ON RANGE 

At long ranges, target strength is practically inde¬ 
pendent of range. Close to the submarine, however, 
the target strength will decrease with range. This 
phenomenon has two causes. First, at very short dis¬ 
tances from the submarine, the submarine reflects 
more like a plane or a cylinder than a sphere, and the 
inverse fourth power law does not apply. In other 
words, the target is not equivalent to a point source, 
and the target strength decreases. This effect de¬ 
pends on the aspect of the submarine and diminishes 
as the range exceeds the maximum radius of curva¬ 
ture of the submarine. 

Secondly, at short ranges the effective portion of 
the sound beam may cover only part of the area ex¬ 
posed by the submarine. Such an effect would be 
expected primarily at beam aspect, since geometric 
foreshortening reduces the area exposed by the sub¬ 
marine at other aspects. For example, a sound beam 
12 degrees wide will not cover the entire length of a 
300-ft submarine, at beam aspect, at ranges less than 
475 yd. However, only those areas on the submarine 
giving rise to nonspecular reflection would be af¬ 
fected, so that if most of the reflection were specular, 
arising in a small area amidships, the target strength 
as measured very close to the submarine from per¬ 
fectly aimed pulses would not depend on the beam 
width. From Figures 1 to 5 in Chapter 22, it appears 
that most of the reflection is concentrated amid¬ 
ships. Thus the effect of beam width, though present, 
is probably not important except possibly at very 
short ranges, as long as nonspecular reflection is 
neglected. 



DEPENDENCE ON RANGE 


403 



0 30 60 90 120 150 180 

BOW BEAM STERN 

ASPECT ANGLE IN DEGREES 

Figure 16. Theoretical dependence of target strength on range for the U570 (HMS/M Graph). 


23.4.1 Theory 

In the theoretical target strength studies described 
in Section 20.5, 6 plans of the Graph were employed 
in calculating target strengths. 16 This submarine has 
a maximum radius of curvature of about 5(30 yd. 
Since target strengths vary with range primarily for 
ranges less than the maximum radius of curvature of 
the submarine, the target strength of the Graph 
would be expected to approach a limiting value as 
the range is increased to 500 or 000 yd. 

Actually, the target strength of the Graph is very 
near this limiting value at ranges beyond only a few 
hundred yards, especially at beam aspect. Target 
strengths have been computed for ranges of 8, 12, 
10, 200, and 1,000 yd on the assumption of a non- 
directional source of sound and are plotted against 
aspect angle in Figure 10. The shorter ranges are the 
distances from the projector to the nearest part of 
the submarine. It is apparent from Figure 10 that the 
variation of target strength with aspect changes 
markedly as the range is reduced from 200 to 10 yd; 
no intermediate ranges were used. 

These calculations were based on a nondirectional 
source; therefore the results neglect the effect of 


limited coverage of the submarine by a directional 
beam at close ranges. Such an effect is not important, 
however, if the reflection is primarily specular and 
comes from a small area amidships, as discussed in 
the preceding section. 

23.4.2 Observations 

Verification of a dependence of target strength on 
range has not been possible in the direct measure¬ 
ments because the transmission loss has not been 
known accurately. Instead, it has been assumed, for 
the ranges used during the target strength measure¬ 
ments— from 200 to 1,000 yd — that the target 
strength at near-beam aspects remains constant. 
Such an assumption is necessary in order to calculate 
the transmission loss; in some of the measurements 
at San Diego, a constant target strength was assumed 
at constant aspect, and the transmission loss was 
computed from a plot of the echo level E plus 40 log 
r against the range r as the range is opened or closed 
(see Section 21.5.1). The most recent measurements 
at San Diego have employed a nondirectional hydro¬ 
phone mounted on the submarine to measure the 
transmission loss; this method, if practical, might 



























404 


SUBMARINE TARGET STRENGTHS 


reduce the usual fluctuations enough to enable an 
evaluation of target strength as a function of range. 

The indirect measurements at MIT, at selected as¬ 
pect angles, and at full-scale ranges of about 250 and 
030 yd, gave about the same values as those at a full- 
scale range of 190 yd. 17 This result is consistent with 
the theoretical computations shown in Figure 16, 
where the maximum change in target strength is only 
about 3 db at beam aspect, as the range changes from 
200 to 1,000 yd. Since for these ranges the target 
strength was found to be independent of range, it 
was apparent that the intensity of the light reflected 
from the models and measured at the receiver varied 
with range at the same rate as that from a sphere, 
rather than that from a cylinder or plane, in other 
words, inversely as the fourth power of the distance. 
The shortest range was approximately twice the 
length of the submarine. It is likely that this relation 
would not hold at much closer ranges where target 
strengths would be expected to depend strongly on 
the range. 

Target strength was found to depend on the range 
in the acoustical model experiments at Mountain 
Lakes, for full-scale ranges of 85, 170, and 250 yd. 
The submarine model behaved as a cylinder, not as a 
sphere or plane. For reflection from a cylinder, the 
echo level should decrease 9 db when the range is 
doubled as long as the range is not much greater than 
the length of the cylinder (see Section 20.4.3); for a 
sphere the same increase in range causes a drop of 
12 db. It was found that the echo level at beam as¬ 
pect actually dropped 8.8 db as the full-scale range 
was doubled, from 85 to 170 yd. Thus the hull at 
beam aspect and at short ranges behaves as a cyl¬ 
inder, as might be expected from its large radius of 
curvature in the horizontal plane. 

In general, the indirect measurements agree with 
the theoretical predictions of the dependence of tar¬ 
get strength on range. This predicted dependence 
should be most marked at ranges less than 200 yd; 
experimentally, it was observed and verified only at 
ranges less than 200 yd. However, too much impor¬ 
tance cannot be attached to these results, since the 
measurements were made only at three particular 
ranges in each of the two indirect measurements, and 
since experimental errors were so large. 

23.5 DEPENDENCE ON PULSE LENGTH 

When short pulses are used instead of continuous 
sound, target strength may depend on the lengths of 


these pulses. Sections 19.3 and 20.7 discussed in an 
elementary way the effect of pulse length on meas¬ 
ured target strengths. For short pulses —- signals 
whose length in the water is less than the length of 
the target in the direction of the sound beam — the 
echo level and therefore the target strength will de¬ 
pend on the signal length. The exact variation of 
target strength with signal length, however, depends 
on whether peak echo intensities or average echo 
intensities are used in computing target strengths. 

23.5.1 Theory 

In most direct measurements of target strength, 
peak amplitudes are measured from the oscillograms 
rather than average amplitudes, because echo pro¬ 
files are so irregular that the average amplitudes over 
the length of the echo would be difficult to measure. 
A simple analysis given in Section 19.3.1 shows that 
for long pulses the average echo intensity is inde¬ 
pendent of signal length, while the echo length varies 
with the signal length. For short pulses on the other 
hand (see Section 19.3.2), the average echo inten¬ 
sity is approximately directly proportional to the 
signal length, while the echo length remains con¬ 
stant. This analysis applies to square-topped pulses 
striking an extended single target. 

It is arbitrary whether peak amplitudes or aver¬ 
age amplitudes are used to compute target strengths. 
Peak amplitudes are easier to measure. In addition, 
most other underwater sound measurements, such 
as those undertaken in the investigation of recogni¬ 
tion differentials, are based on peak amplitudes. It 
may be, however, that the ear, or the sound range 
recorder, or other detection devices may respond to 
the average echo intensity instead of the peak echo 
intensity, or to the total energy in the echo. There¬ 
fore a comparison of average and peak echo ampli¬ 
tudes and their variation with signal length might be 
a profitable study. 

The change of average and peak echo intensities 
with pulse length has been investigated theoreti¬ 
cally, 18 as described in Section 21.6.4. First it is as¬ 
sumed that the length of each individual peak in an 
echo is approximately equal to the signal length. 
Then it is assumed that these peaks are statistically 
independent, or distributed at random throughout 
the length of the echo. By assuming that the echo 
is essentially a group of rectangular peaks, each of 
which follows the Rayleigh distribution for succes¬ 
sive echoes and each of which is independent of the 



DEPENDENCE ON PULSE LENGTH 


405 




Figure 17. Dependence of target strength on signal 
length for an S-boat at 24 kc. 


others, the change in the peak echo intensity, aver¬ 
aged over many echoes, turns out to be considerably 
less than the change in mean echo intensity, aver¬ 
aged over the length of the echo and then over many 
echoes, for the same change in pulse length. 

For example, it is shown that, for a uniformly re¬ 
flecting target whose length is 300 ft in the direction 
of the sound beam, corresponding roughly to a fleet- 
type submarine at bow or stern aspects, the average 
peak echo intensity will decrease only 3 db and the 
average mean echo intensity will decrease 5 db, when 
the signal length is reduced from 30 to 10 msec. The 
assumptions on which this analysis is based are so 
idealized that the exact numerical results are prob¬ 
ably not very significant; nevertheless, the general 
conclusion that the peak amplitude changes less 
rapidly with the signal length than the average 
amplitude seems fairly well established theoretically. 




O 0.2 0.4 0.6 0.8 

BEAM 

COS 9 

Figure 18. Dependence of target strength on signal 
length for a fleet-type submarine at 24 kc. 


1.0 

BOW 

STERN 



aspect angle in degrees 


Figure 19. Dependence of relative target strength on 
signal length at different aspects for an S-class sub¬ 
marine at 60 kc. 

































































SUBMARINE TARGET STRENGTHS 


400 



Figure 20. Dependence of target strength on signal length for an S-boat at 24 kc. 


23.3.2 Observations 

Since evidence showing a dependence of target 
strength on pulse length could be found only in the 
direct measurements, these measurements have been 
examined and analyzed from this point of view. 
Different signal lengths have been employed in 
direct target strength measurements at San Diego, 
where pulses from 0.5 to 200 msec long have been 
used. 

Peak echoes front 5-msee signals were found to 
average about 4 db lower than echoes from 33-msec 
signals, according to early runs at San Diego on an 


S-class submarine, using JK gear at a frequency of 
24 kc. 19 Further studies showed a minimum depend¬ 
ence of target strength on pulse length at beam 
aspects, and a maximum, nearly linear, dependence 
at bow and quarter aspects. 20 Later measurements, 
however, reported no very significant dependence of 
target strength on signal length for signal lengths of 
10, 30, and 100 msec. 3 These measurements are illus¬ 
trated in Figures 17 and 18 for an S-boat and a fleet- 
type submarine respectively; the difference in target 
strengths between 100- and 30-msec signals and 
100- and 10-msec signals were greatest at aspects 
near the bow and stern and are plotted against the 































DEPENDENCE ON PULSE LENGTH 


407 


TARGET STRENGTH IN DECIBELS 
BOW 



STERN 


Figure 21. Dependence of target strength on signal length for an S-boat at 60 kc. 


cosine of the aspect angle. The mean curve connect¬ 
ing the mean points is also indicated. However, this 
dependence on pulse length is relatively small and 
not very reliable in view of the large scatter. 

Measurements were made at a frequency of 60 kc 
on another S-boat at creeping speed and a keel depth 
of 100 ft, with signal lengths of 1, 10, and 30 msec. 21 
A significant variation in relative target strength 
with signal length was observed and plotted in Fig¬ 
ure 19 for bow, beam, and stern aspects where each 
point at beam aspect represents about 40 echoes, at 
stern aspect about 20 echoes, and at bow aspect only 
a few. Surprisingly, a large variation with pulse 


length is found at beam aspect; here theory would 
predict a minimum dependence, since the echoes for 
the most part accurately reproduce the pulses. An 
attenuation coefficient of 20 db per kyd was assumed 
in calculating the transmission loss; ranges averaged 
about 300 yd. 

Figures 20 and 21 show target strength as a func¬ 
tion of aspect angle for three different signal lengths, 
1, 10, and 30 msec, for a submarine of the S class at 
24 and 60 kc. A definite dependence on signal length 
is evident from an examination of the three curves in 
each figure, although the actual target strength val¬ 
ues are rather low. The dependence on aspect angle 


























408 


SUBMARINE TARGET STRENGTHS 


TARGET strength in decibels 



Figure 22. Dependence of target strength on frequency (San Diego). 


is somewhat obscured by the fluctuations encoun¬ 
tered during these particular runs. 

In general, target strength depends on the signal 
length for short signals althqugh it varies less rapidly 
than the signal length, or rather, less rapidly than 10 
log t, where r is the signal length; a decrease in tar¬ 
get strength is most marked at signal lengths less 
than 10 msec and at aspects away from the beam. 

23.6 DEPENDENCE ON FREQUENCY 

Both sonic and supersonic frequencies have been 
employed in echo ranging. Lower frequencies are de¬ 


sirable from the point of view of transmission, since 
the transmission loss increases with frequency. On 
the other hand, since high-frequency sound is more 
directive, the bearings of targets may be located more 
accurately at high frequencies than at low frequen¬ 
cies. As a result, choosing a frequency for echo rang¬ 
ing is always a compromise between these two char¬ 
acteristics. 

Target strengths have been predicted and meas¬ 
ured directly at frequencies between 12 and 60 kc. 
Theoretical calculations have been based on a fre¬ 
quency of 25 kc. Most of the direct measurements 
have been made at 24 kc, since this frequency is 































DEPENDENCE ON FREQUENCY 


409 


standard for most Navy sonar gear, although some 
measurements have been made at frequencies of 12, 
18, 45, and GO kc. The indirect measurements used 
scale models; at Mountain Lakes, with a 1:60 scale 
model of the Graph, a frequency of 1,565 kc was em¬ 
ployed to simulate an echo-ranging frequency of 
26 kc. At MIT visible light was used, and the cor¬ 
responding full-scale frequency was very much 
higher than for any of the other measurements. 

23.6.1 Theory 

The target strength of a submarine depends on 
frequency according to equation (36) in Chapter 20, 
especially if nonspecular reflection contributes ap¬ 
preciably to the target strength. Specular reflection 
depends only slightly on frequency, as described in 
Section 20.4. Beam echoes result largely from specu¬ 
lar reflection, as pointed out in Section 23.8.1. Thus 
the variation of target strength with frequency at 
beam aspects may be expected to be slight. 

At off-beam aspects, however, specular reflection 
is much less important and nonspecular reflection 
may become appreciable; this effect is discussed in 
Section 23.8.2. At these aspects, the whole sub¬ 
marine appears to scatter sound. If reflections from 
the superstructure or exterior protuberances on the 
submarine, such as rails, guns, and periscopes, con¬ 
tribute appreciably to the target strength, the target 
strength may depend on frequency as long as the 
dimensions of these scatterers are of the same order 
of magnitude as the wavelength. Section 20.5 de¬ 
scribes the origins of nonspecular reflection. 

23.6.2 Direct Measurements 

No reliable direct measurements substantiate this 
expected dependence of target strength on frequency. 
Figure 22 shows target strength as a function of as¬ 
pect angle plotted for frequencies of 24 and 60 kc for 
a fleet-type submarine at San Diego at signal lengths 
of 10, 30, and 100 msec. Each point represents the 
average of all observations in a 30-degree sector 
centered at the point indicated. 

A clear-cut dependence of target strength on fre¬ 
quency is apparent in this illustration, as well as in 
Figures 20 and 21. Figure 21 gives quite reasonable 
values for the target strength at 60 kc, but Figure 20 
shows values about 10 db lower, for a frequency of 
24 kc; thus the dependence on frequency is still evi¬ 
dent. It is unlikely, however, that the increase in tar¬ 
get strength with frequency would be not only so 


great but also so nearly uniform at all aspect angles, 
as Figure 22 shows. Furthermore, the measurements 
cannot be relied on for two reasons: the transmission 
loss was not known but estimated, and the calibra¬ 
tion of the 60-kc gear was less reliable than the 24-ke 
gear calibration. The estimated attenuation co¬ 
efficient of 20 db per kyd used for these computations 
is larger than that measured elsewhere and is per¬ 
haps excessive, since attenuation coefficients of only 
10 db per kyd at 60 kc were measured at Fort 
Lauderdale. However, correcting the high San Diego 
target strengths at 60 kc by reducing the attenuation 
coefficient from 20 to 10 db still results in values 
greater than those obtained elsewhere. The remain¬ 
ing discrepancy may perhaps be attributed to faulty 
calibration of the gear, described in Section 21.4, 
since this discrepancy is systematic and apparently 
independent of aspect angle. 

In addition, the high target strengths measured at 
60 kc at San Diego do not seem substantiated by 
target strength measurements at that frequency at 
Fort Lauderdale. These measurements gave a target 
strength of 25 db for a fleet-type submarine at beam 
aspect at 60 kc, assuming an attenuation coefficient 
of 12 db per kyd. An assumption of an attenuation 
coefficient of 20 db per kyd would raise this to only 
33 db, compared with the maximum value of 44 db 
recorded at San Diego for the target strength of a 
fleet-type submarine. Furthermore, wake echoes 
measured with the same equipment at San Diego 
and described in Section 33.4.2 were found to be 
much higher at 60 kc than at 24 kc, contrary to 
theoretical expectations. These results support the 
suggestion that calibration errors are responsible for 
the high values obtained at San Diego. However, in 
view of the many uncertainties in this subject, the 
possibility that submarine target strengths are sys¬ 
tematically some 10 db higher at 60 kc than at 24 kc 
cannot be entirely ruled out, even though there is 
little if any theoretical expectation of such a varia¬ 
tion. 

No difference was apparent between the measure¬ 
ments at 12 kc and 24 kc made by Woods Hole ob¬ 
servers; 10 both target strength-aspect curves were 
very similar. However, the target strengths reported 
are so much larger than all other measurements else¬ 
where that calibration errors were probably present. 
Therefore, since the calibration at 12 kc was quite 
different from that at 24 kc, and since all the values 
seem very uncertain, the lack of any frequency de¬ 
pendence cannot be considered significant. 



410 


SUBMARINE TARGET STRENGTHS 


23.6.3 Indirect Measurements 

Visible light from a motion picture projection bulb 
was used in the optical measurements at MIT. Since 
the frequency, or the band of frequencies, was not 
properly scaled to correspond with usual echo¬ 
ranging frequencies, it was not practical to investi¬ 
gate the dependence of target strength on frequency. 
However, since improperly scaled light was used to 
give results for comparison with direct and other in¬ 
direct measurements, four frequency effects should 
be remembered in interpreting the results of the op¬ 
tical measurements. 12 

First, at certain aspects when the insonified sur¬ 
face of the actual submarine subtends only a few 
Fresnel zones at 24 kc, the surface of the model il¬ 
luminated by visible light subtends many zones, 
since the wavelength of the light was much shorter 
compared with the dimensions of the model than was 
the wavelength of the sound used in echo ranging 
compared to the dimensions of an actual submarine. 

As a result, for the optical measurements, the ex¬ 
pressions for the target strength due to the effects of 
a group of Fresnel zones approached their asymp¬ 
totic values, especially for surfaces with at least one 
large radius of curvature, such as cylinders and 
planes. On submarines this effect might apply to the 
conning tower, keel, and top deck, so that in the 
optical measurements the effect of the conning tower, 
and the effect of the deck at an altitude angle of 90 
degrees, might be overemphasized. 

Secondly, nonspecular reflection is too small by a 
factor of the square root of the ratio of the actual 
wavelength used to the properly scaled wavelength. 
In the optical measurements, this factor is about 45, 
or about 17 db. In other words, nonspecular reflec¬ 
tion measured optically is about 17 db too low. 
This factor may account for the very low target 
strengths obtained optically at off-beam aspects 
where nonspecular reflection may be more important. 

Thirdly, diffuse reflection or scattering may be too 
great optically because the wavelength may be much 
smaller than the surface irregularities. An attempt 
was made to minimize this error by using glossy 
black surfaces. 

Finally, where two or more specular reflections 
occur, the light beams do not interfere, as sound 
beams do, because they are incoherent. Since the 
incoherent sum is an average of the interference pat¬ 
tern, this sum may actually be more interesting than 
the detailed interference pattern itself, as the aver¬ 


age is more significant, in most applications to prac¬ 
tical echo ranging at sea, than the exact pattern. 

Thus the results of the optical measurements, 
though suggestive of what might be encountered in 
practical echo ranging, cannot be compared directly 
with the other measurements unless these effects of 
the wavelength are considered and accounted for. 

In the acoustical measurements at Mountain 
Lakes, no long-term systematic variation with fre¬ 
quency was observed for full-scale frequencies from 
about 1 to 35 kc. Figure 23 shows relative echo level 
plotted against frequency for the Mountain Lakes 
measurements on the Graph at beam aspect, at a 
full-scale range of about 15 yd. The peaks and dips 
evident in this illustration are largely the result of 
interference phenomena arising from two specular 
reflections from the hull and conning tower, as the 
frequency is changed, and of the response character¬ 
istics of the system. Interference phenomena result¬ 
ing from multiple reflections from several surfaces 
on the submarine are clearly shown in Figure 24, 
where the relative beam target strength is plotted 
against altitude angle for a full-scale frequency of 
26 kc. The nearly smooth curves at altitudes of 0 and 
90 degrees result from direct specular reflection from 
the deck and hull respectively. The intricate interfer¬ 
ence pattern at altitude angles between 10 and 50 
degrees and 270 and 350 degrees results from path 
differences in the sound doubly reflected from the 
hull and from the blister tank; similar patterns are 
evident for sound reflected from the bottom of the 
submarine model. 

23.7 OCEANOGRAPHIC CONDITIONS 

Target strength measures the reflecting character¬ 
istics of a target and is computed from the echo level, 
source level, and transmission loss from equation (6) 
in Chapter 19. Since it depends only on the target 
itself, it is theoretically independent of the medium 
and its characteristics, independent of the transmis¬ 
sion characteristics of sound in water, and therefore 
independent of oceanographic conditions insofar as 
they affect transmission. 

In practice, however, reported target strengths 
have been found to depend markedly on the prevail¬ 
ing oceanographic conditions in cases where the 
transmission loss has not been accurately known. 
Since target strength is computed from the echo 
level, source level, and transmission loss, improper 
appraisal of the transmission loss will appear as an 
error in the target strength values reported. 



RELATIVE TARGET STRENGTH AT BEAM ASPECT 
IN DECIBELS 



Figure 23. Dependence of target strength on frequency (Mountain Lakes). 



Figure 24. Double reflection and interference (Mountain Lakes). 






































































































































































































































































































































































































































































































OCEANOGRAPHIC CONDITIONS 


411 


Section 21.5 introduced the concept of an attenua¬ 
tion coefficient to describe more accurately the trans¬ 
mission loss. This attenuation coefficient varies both 
with the oceanographic conditions and the frequency 
of the echo-ranging sound. From a quantity of trans¬ 
mission data at 24 kc two empirical formulas were 
suggested for estimating the attenuation coeffi¬ 
cient, 22 equations (5) and (6) in Chapter 21. They 
are 


for the hydrophone above the thermoeline, and 

260 

a = 4.5 +- — (2) 

for the hydrophone below the thermoeline, where a 
is the attenuation coefficient in decibels per kiloyard 
and D the depth of the thermoeline in feet. The prob¬ 
able error of this estimate is about 2 db per kyd. 


measurements at 24 kc on different S-boats, calcu¬ 
lated by assuming an attenuation coefficient of 5 db 
per kyd, showed such deviations that a marked de¬ 
pendence on the particular submarine used was sug¬ 
gested. 23 The particular submarines used are desig¬ 
nated in the first column of Table 4, while the target 
strengths, varying from 7 to 25 db, are reproduced 
in the second column. Investigation of the oceano¬ 
graphic conditions prevailing during the different 
runs showed an unmistakable correlation between 
the target strengths and the thermoeline depths; 
when the thermoeline depth was only 18 ft, the com¬ 
puted target strength was only 7.3 db, while it rose 
to 25 db for a thermoeline 160 ft deep. 

Accordingly, equations (1) and (2) were used to 
calculate new and presumably more reliable attenua¬ 
tion coefficients, from the thermoeline depths, listed 
in the third column, and the depths of the submarines 
during the runs, in the fourth column of Table 4. 


Table 4. Dependence of reported target strengths on attenuation coefficient. 


S-boat 

Reported beam 
target strength 
in decibels 

Depth to 
thermoeline 
in feet 

Depth of 
submarine 
in feet 

Computed 
attenuation 
coefficient 
in decibels 
per kiloyard 

Range 
in yards 

Correction 
in decibels 

Computed beam 
target strength 
in decibels 

USS S-28 (SSI33) 

18* 

85 

45 

5.5 

350-520 

+0.5 

18.5 

USS S-40 (SSI45) 

25* 

160 

90 

4.6 

300-400 

-0.3 

24.7 

USS S-23 (SS128) 

13.7* 

50 

90 

9.7 

350-500 

+4.7 

18.4 

USS S-23 (SSI28) 

15.4* 

75 

100 

5.8 

330 

+0.3 

15.7 

USS S-33 (SSI38) 

7.3* 

18 

90 

19.0 

415 

+ 11.6 

18.9 

USSS-31 (SSI 36) 

22.3t 

100 

95 

5.2 

440 

0.0 

22.3 

Mean beam 
target strength 
and 

standard deviation 
in decibels 

16.9 ± 4.8 






19.7 ± 2.5 


* Assumed attenuation coefficient, 5 db per kyd. 
t Measured attenuation coefficient, 5.3 db per kyd. 


23.7.1 Effect oil Measurements 

Direct measurements of the transmission loss dur¬ 
ing target strength runs on submarines have been 
difficult and generally unsuccessful, as described in 
Section 21.5.2. As a result, it has been customary at 
San Diego to compute the transmission loss from an 
estimated attenuation coefficient at the particular 
frequency employed, usually 5 db per kyd at 24 kc 
and 20 db per kyd at 60 kc. 

Target strengths computed in this way show enor¬ 
mous differences. For example, various series of 


These new values appear in the fifth column, the ap¬ 
propriate ranges in the sixth column, in the seventh 
column the corrections resulting from the assumed 
value of 5 db per kyd of sound travel, not range — 
assuming the maximum range from the sixth column 
— and the new' target strengths in the last column. 
Two results are noteworthy: the standard deviation 
is reduced by a factor of almost two from 4.8 db to 
2.5 db, and the mean beam target strength for an 
S-boat is raised from 16.9 db to 19.7 db. The new T 
value agrees more closely with other measurements. 

At 60 kc few'er data are available. An assumption 
















412 


SUBMARINE TARGET STRENGTHS 



QUARTER 



note: first echo is a surface wake quarter 


Figure 25. Oscillograms of submarine echoes (S-class submarine). 








'i '.III,jin l , !i';.'|i||||i | i ( ; l !ll'l„l;'l| 


STRUCTURE AND ORIGIN OF ECHOES 


413 


RANGE 


ASPECT ANGLE 
IN DEGREES 


PORT 

BEAM 

270 







-BOW 

0 



of 20 db per kycl for the attenuation coefficient, al¬ 
though assumed in the San Diego target strength 
computations, is not substantiated by tests made at 
Fort Lauderdale which gave results of 9 to 10 db per 
kyd. These latter measurements were made by plot¬ 
ting the echo level, corrected for inverse square loss, 
against the range, as the range continuously de¬ 
creased from about 600 to 150 yd; the submarine was 
at stern aspect throughout the run. This discrepancy 
was mentioned in Section 23.6.2. The high value as¬ 
sumed for the attenuation coefficient at San Diego 
was suggested by early transmission measurements; 
in these tests, the presence of shallow thermoclines off 
the coast of California was partly responsible for the 
high values measured. For deep, mixed water, lower 
values are more common 24 (see Chapter 5). 



Figure 27. Oscillograms of submarine echoes at beam 
aspect for 10-millisecond pulses. 



Figure 26. Sound range recorder records of submarine 
echoes from 30-millisecond pulses. 


23.8 STRUCTURE AND ORIGIN OF 
ECHOES 

Most target strengths have been measured from 
echoes recorded oscillographically on 35-mm motion 
picture film. At San Diego, this film was run past the 
oscilloscope at a speed sufficiently high to record the 
detail of each echo; depending on the signal length 
and the exact film speed, these echoes may be from 
0.2 to 3 or 4 cm long. Accordingly, these echoes have 
been carefully studied in an attempt to formulate 










414 


SUBMARINE TARGET STRENGTHS 



BOW 


180 


STERN 



QUARTER 


QUARTER 


QUARTER 

Figure 28. Oscillograms of submarine echoes at off-beam aspects for 10-millisecond pulses. 













STRUCTURE AND ORIGIN OF ECHOES 


415 


more precise conclusions regarding the process of 
reflection of sound from submarines. 

Examination of hundreds of oscillograms of echoes 
at San Diego has made possible a separation of 
echoes into two classes, beam and off-beam, as illus¬ 
trated in Figure 25. Each class shows its own char¬ 
acteristics and peculiarities, on the basis of which 
tentative explanations of reflection phenomena have 
been made. These two types of echoes produce such 
different traces on the sound range recorder that the 
appearance of these traces is used tactically to esti¬ 
mate the aspect of the target. Typical sound range 
recorder traces are illustrated in Figure 2(3. 


23.8.1 Beam Echoes 

Beam echoes are always stronger, on the average, 
than echoes at any other aspect, both according to 
the theoretical calculations and the direct and indi¬ 
rect measurements. There are four lines of evidence 
which indicate that reflection is specular and arises 
primarily from the hull of the submarine. 

First, theory predicts strong specular reflection at 
beam aspect. 6 The theoretical values derived assum¬ 
ing only specular reflection are in excellent agree¬ 
ment with other values measured directly and in¬ 
directly. The effect of the conning tower appears to 
be negligible for the U570 at beam aspect, since it 
contributes only 0.2 db to the target strength at long 
ranges. 

Secondly, the oscillograms of beam echoes are 
clear cut and closely resemble the square-topped 
pulses sent out; further examples are illustrated in 
Figure 27, which shows oscillograms of three succes¬ 
sive echoes from a submarine at beam aspect. These 
beam echoes contrast sharply with off-beam echoes, 
which are illustrated in Figure 28. Occasionally, 
beam echoes show very sharp and narrow peaks at 
either end, or a short “tail” of lower intensity, which 
are attributed to two different types of surface reflec¬ 
tions described in Section 21.5.4. A typical sound 
range recorder record of the double echoes at beam 
aspect is illustrated in Figure 29. Since beam echoes 
usually equal the signal in length, for signals 10 or 
more milliseconds long, the effective reflecting sur¬ 
face does not appear to be much extended in the 
direction of the sound beam; typical oscillograms of 
beam echoes are illustrated in Figure 30. In other 
words, the relatively flat area on the hull of the sub¬ 
marine is responsible for almost all the energy in the 
echoes received at beam aspect. This same specular 


RANGE IN YARDS 


800 
I I 


-t-.Jg ■ T.T J 

b gt > 5 

ip-.'I*; 




'•i 


25 


!|j '*'• f: 

152:55 <•' - ~L 


-2>* 2 
* 


W 


v'JLZO 


sr>: 


5;-r- •*'< 


Figure 29. Double echoes recorded on the sound 
range recorder. 

reflection may be inferred from Figure 24 where a 
fine interference pattern is conspicuously absent at 
beam aspect for altitude angles in the neighborhood 
of 0 degree. 

Thirdly, optical measurements on a model of the 
Sand Lance as well as both optical and acoustical 
measurements on models of the U570 gave identical 
target strength at beam aspect with and without the 
conning tower, over a sector of about 20 degrees, as 
Figures 31, 32, and 33 show. The conning tower, 
although important at other aspects, contributes 
little to reflection at beam aspects. 

Fourthly, the importance of hull reflections is 
evident in Figures 1 through 5 of Chapter 22. Al¬ 
though these photographs refer only to optical illu¬ 
mination of the model and may not apply perfectly to 
the reflection of supersonic sound from submerged 
submarines, they may be representative of what hap¬ 
pens acoustically. 

However, measurements made with short pulses 
at beam aspect show a detailed echo structure which 
suggests that not all the reflected sound comes from 
the submarine hull or ballast tanks. Measurements on 






SUBMARINE TARGET STRENGTHS 


416 


- 

1/2-MILLISECOND PULSE ECHO 



SUCCESSIVE ECHOES 



10-MILLISECOND PULSE 


ECHO 


30-MILLISECOND PULSE ECHO 



100 -MILLISECOND PULSE ECHO 



Figure 30. Detailed oscillograms of submarine echoes at beam aspect. 










STRUCTURE AND ORIGIN OF ECHOES 


417 


BOW 


BOW 




BEAMj 


STERN 


Figure 31. Effect of conning tower on optical meas¬ 
urements on USS Sand Lance (SS381). 


Figure 32. Effect of conning tower on optical meas¬ 
urements on U570 (HMS/M Graph). 


an S-boat at beam aspect for signals from 0.5 msec 
long resulted in echo oscillograms showing two dis¬ 
crete “blobs” of about equal mean amplitudes with a 
range separation of about 4 yd. 25 ' 26 Although the finer 
details of the echo envelopes and the relative values 
of the peak amplitudes did not repeat from echo to 
echo, the main features consistently suggested the 
presence of two distinct reflecting surfaces on the 
submarine at beam aspect. For signals longer than 
4 yd (5 msec), the echo envelopes were almost al¬ 
ways resolved into three distinct segments, with the 
central portion corresponding to the overlap or addi¬ 
tion of the two primary echoes found for shorter sig¬ 
nals. The amplitude of this central portion presum¬ 
ably varied according to the initial phase difference 
and amplitude of the two component signals; the 
difference between the phases changed little during 
the period of reflection. Typical oscillograms are il¬ 


lustrated in Figure 30; the weak echo following the 
main one is sound reflected from the submarine up to 
the surface, back to the submarine and then back to 
the projector, as discussed in Section 21.5.4. Because 
the individual echo components in Figure 30 appear 
to be coherent, at beam aspects the echo is probably 
specular. Nonspecular or diffuse reflection is appar¬ 
ently unimportant at these aspects, although other 
surfaces besides the hull and ballast tanks may con¬ 
tribute to the echo. 

23.8.2 Off-Beam Echoes 

Echoes from aspects other than beam are generally 
very different from beam echoes. Not only is the echo 
weaker, but it is also less well defined; examples are 
shown in Figures 25 and 28. If a system of high re¬ 
solving power is used, such as the oscilloscope and 





















418 


SUBMARINE TARGET STRENGTHS 


BOW 



STERN 

Figure 33. Effect of conning tower on acoustical 

measurements on U570 (HMS/M Graph). 

high-speed camera at San Diego, the echoes appear 
to be a group of fine spikes or peaks, although some 
evidence points to peaks which are found in the same 
places in successive echoes. A more complete discus¬ 
sion of the detailed structure of off-beam echoes and 
their origin is postponed to the next section. Here the 
more general features of off-beam echoes are dis¬ 
cussed. 

The beginning and end of an echo at off-beam 
aspect are not clearly defined; usually the amplitude 
builds up and dies away gradually, blending into the 
background at either end, so that precise measure¬ 
ment of the echo length is impossible. However, an 
examination of off-beam oscillograms shows that 
these echoes are longer than the signals and there¬ 
fore suggests an extended target. 21 

If the entire length of the submarine is effective in 


reflecting sound, as seems indicated for off-beam 
echoes, the lengths of these off-beam echoes should 
vary with the aspect angle, depending on the length 
of the submarine in the direction of the sound beam. 
In other words, if a submarine is an extended target 
and scatters sound throughout its length, the length 
of the echoes which it returns should depend on the 
aspect which it presents to the sound beam. 

Accordingly, the lengths of these off-beam echoes, 
diminished by the length of the signal used, have been 
measured or estimated as accurately as possible, 
then plotted against the cosine of the aspect angle 
which accounts for the foreshortening of the sub¬ 
marine. 27 Figure 34 illustrates the results of this 
analysis, where the broken curve connects the meas¬ 
ured points, and the solid curve is a polar plot of the 
cosine of the aspect angle, modified at bow aspect to 
account for the “shadow” which the forward section 
casts on the stern section. A similar dip is included at 
stern aspect, where the after part of the submarine 
shadows the forward part. 

The maximum elongation — echo length minus 
signal length — was found to occur at quarter as¬ 
pects, roughly 15 degrees from the bow and stern 
on either side, and amounted to about 85 yd. 26 The 
actual length in the direction of the sound beam of a 
fleet-type submarine 300 ft long, at aspects 15 de¬ 
grees from bow and stern, is about 96 yd, which con¬ 
firms the suggestion that the entire surface of the 
submarine scatters sound. At an aspect angle of 135 
degrees, when the target had an extension of 49 yd 
in the direction of the sound beam, the elongation 
amounted to about 38 yd. 

At bow aspect, this elongation was reduced to 
50 yd. This reduction is attributed to the shadow 
cast by the forward section on the after section. A 
similar drop, however, was not observed at stern 
aspect. 

These elongation phenomena are apparently inde¬ 
pendent of signal length and echo-ranging frequency. 
Apparently they are the result of scattering from the 
entire length of the submarine, instead of reflection 
from only one or two major surfaces such as the 
conning tower or screws. They suggest that in addi¬ 
tion to specular reflection from the hull, nonspecular 
reflection or diffuse scattering also occurs, especially 
at aspects away from the beam. The exact mechanism 
by which sound is reflected from the entire length of 
the submarine is unknown. 

Similar elongation phenomena were analyzed by 
British observers in an effort to determine the origin 










STRUCTURE AND ORIGIN OF ECHOES 


419 



I 80 
STERN 


90 

BEAM 


Figure 34. Dependence of echo elongation on aspect angle (San Diego). 



















420 


SUBMARINE TARGET STRENGTHS 




















— 

- - 

• ^ 



-PROJECTED LENGTH OF SUBMARINE IN DIRECTION OF 
, SOUND BEAM (DIFFERENCE IN RANGE BETWEEN 



- - 






s/ 

\ 







/ 

s 










-V 

\ 





/ 

/ 






• 

a m 






\ 





/ 

/ 






— 

• 

• i 

• 

• 

• 

» 

• 

• • 

» 

\ 

\ 



-/ 

/ 

/ 



i 

• 

•v 

• 

• • 

• 

• • 

•• 

• 

vf 

• 






• • 
• 

• 

*• • 

\ 

• • 



/ 

• • 

• 

•i 

• • 

• 

• 


• 

• 









• 

'w* - 

• 

• • 

• 

• 

^ •• 

• • 

• 

• 






0 30 60 90 120 150 180 


BOW 


BEAM 

ASPECT ANGLE IN OEGREES 


STERN 


Figure 35. Echo elongation and projected length of submarine as a function of aspect angle. 


of the nearest echo, but the hypothesis that the en¬ 
tire submarine reflected sound was not confirmed. 28 
The elongation, plotted in Figure 35, amounted to 
only about half the calculated exposed lengths of the 
submarine, after the pulse lengths had been sub¬ 
tracted from the echo lengths. Similar results have 
also been obtained in this country using the same 
technique. It may be pointed out, however, that a 
sound range recorder, not an oscilloscope, was used 
in these experiments, and that records from a sound 
range recorder might be expected not to show the 
weaker tail part of the echo. 

23.8.3 Source of Echoes 

Beam echoes originate in large part at the hvdl of 
the submarine, as described in Section 23.8.1, with 
some additional contribution possible from the hull 
and bilge keel. The echoes are nearly square-topped 
and result from simple specular reflection from only 
one or two surfaces on the submarine. 

Off-beam echoes, however, apparently come from 
all parts of the submarine. Oscillograms of these 
echoes are detailed and show a fine microstructure of 
peaks and valleys, somewhat similar to reverbera¬ 
tion, especially for short pulses. Since study of the 
elongation phenomena suggests that echoes are re¬ 
turned from most of the submarine, various peaks in 
the detailed structure of an echo might be correlated 
with discrete reflecting surfaces on the outside of the 
submarine. Only short signals could be used, how¬ 


ever; otherwise the signals from individual reflectors 
on the submarine might overlap. 

Accordingly a series of echo oscillograms from sig¬ 
nals approximately 0.5 msec (0.4 yd) long were 
studied at San Diego. 27 The target was a submarine 
of the S class at quarter aspect, 135 degrees from the 
bow. The echoes were recorded oseillographically, as 
usual, but the film was run at a speed of about 13 in. 
per sec, five times faster than normally. This high 
speed lengthened each echo and permitted better 
resolution of the echo structure. 

Each echo analyzed consisted of a number of sharp 
spines, usually between twenty and fifty, which rose 
clearly above a fuzzy background. The envelope of 
these spines was roughly cigar-shaped while the en¬ 
velopes of the less intense parts of the echo peaks 
were similarly shaped but only about half the ampli¬ 
tude of the spine structure. The distribution of these 
spines appeared to be random, and no peaks or 
groups of peaks could be definitely correlated with 
individual reflecting surfaces, such as the conning 
tower or ballast tanks. Thus the peaks may be more 
the result of constructive interference of sound scat¬ 
tered at random from the entire submarine, than of 
strong reflections from discrete surfaces on the sub¬ 
marine. 

Studies of other short-pulse echoes obtained at 
other aspects from various submarines usually yield 
somewhat similar results. The echo almost always 
consists of a succession of peaks rising above the 
background. These peaks, however, do not always 






































STRUCTURE AND ORIGIN OF ECHOES 


421 



occur at the same place in successive echoes; rather, 
they usually appear to be distributed unsystemati¬ 
cally though generally nearer the center of the echo 
than either end. 

In some cases, however, repeatable peaks seem to 
be present in submarine echoes. A series of nine con¬ 
secutive echoes from 5-msec signals at 00 kc are repro¬ 
duced in Figure 36 and show two separate peaks or 
groups of peaks at the same places in each echo. In 
these measurements the submarine aspect was held 
nearly constant at about 330 degrees. Thus no defi¬ 
nite conclusions can be drawn at the present time as 
to how often an echo peak will reproduce itself. It is 
therefore uncertain whether these peaks represent 
highlights on the submarine or random interference 
between several reflected sound waves. 

In general, the process of reflection of sound from a 
submerged submarine at off-beam aspects is still im¬ 
perfectly understood. The entire submarine appears 
to contribute to the reflected sound, yet specific, 
repeatable highlights have not been observed in most 
examinations of echo oscillograms. It is difficult to 
understand how nonspecular reflection from the sub¬ 
marine hull or from protuberances and fixtures on the 
outside of the submarine can account tor these 
echoes. Until the origin of these off-beam echoes from 
actual submarines is satisfactorily explained, the ap¬ 
plicability to actual echo ranging of the results ob¬ 
tained with the indirect optical and acoustical tests 
is open to question. 


Figure 36. Repeatable peaks in submarine echoes. 






Chapter 24 


SURFACE VESSEL TARGET STRENGTHS 


M uch less is known about surface vessel tar¬ 
get strengths than about submarine target 
strengths. Few measurements have been made of the 
sound-reflecting characteristics of ships, and much 
of the available information has been extracted from 
experiments where the investigation of the target 
strength was only incidental to other studies. No 
mathematical analyses or measurements on scale 
models have been attempted, so that all target 
strengths reported here are the results of direct 
measurements. 

Experimental conditions have been far from con¬ 
trolled during these measurements. Ship speed, 
course, range, and especially aspect angle have been 
difficult either to estimate accurately or to maintain 
closely. Many of the tests were made completely at 
random on vessels happening to pass in the vicinity. 
Various types of ships served as targets — destroy¬ 
ers, freighters, tankers, coal colliers, transports, and 
Liberty ships — with the result that although many 
measurements were made, the data on each ship are 
too scanty to afford a comparison between different 
ships. Many variables might have significantly af¬ 
fected the measured echo levels — ship speed, length, 
width, draft, hull curvature, course, range, aspect 
angle, sea state, wind force, temperature gradients —- 
so many that a clear-cut separation of variables is 
out of the question. Furthermore, the results are so 
few in number, compared with other underwater 
sound measurements, and the scatter of values is so 
wide, that only the most tentative and general con¬ 
clusions may be suggested at the present time. 

One of the most important generalizations that 
may be suggested is the difference between the re¬ 
flecting properties of moving vessels and still vessels. 
Ships under way are known to entrain air along their 
sides as thej^ move through the water (see Section 
27.3). With still vessels, on the other hand, en¬ 
trained air seems less likely. Since small air bubbles 
are extremely efficient scatterers of sound, it is rea¬ 
sonable to expect that sound striking a moving vessel 


might be scattered diffusely bv the air bubbles along 
the sides, just as sound is scattered by the wake laid 
by the ship. A still vessel, however, might be ex¬ 
pected to reflect sound specularly. Such an hypothe¬ 
sis seems to bring some coherence into the observed 
data. Therefore, it is largely from this point of view 
that surface vessel target strengths are examined, in 
this chapter, as a function of aspect angle, range, 
ship speed, ship type, pulse length, and frequency. 

24.1 TECHNIQUES OF SAN DIEGO 
MEASUREMENTS 

Surface vessel target strengths have been measured 
by only two groups, the University of California 
Division of War Research at the U. S. Navy Radio 
and Sound Laboratory, San Diego, California 
[UCDWR], and Bell Telephone Laboratories, New 
York, New York [BTL]. 1 The measurements off San 
Diego were made from November 12 to 17, 1943, 
during a program investigating the acoustical prop¬ 
erties of wakes laid by ships at various speeds. 2 

During these tests the USS Jasper (PYcl3) echo 
ranged on two flush-decked World War I destroyers, 
the USS Crane (DD109) and the USS Lamberton 
(DMS2, ex-DD119), which followed straight courses 
at speeds of 10, 15, and 20 knots in deep water. A 
standard Navy JK transducer was used, sending out 
pulses 10 msec long at a frequency of 24 kc. 

The Jasper ranged on the destroyer as it ap¬ 
proached; then, just as the beam of the destroyer 
passed, the Jasper began to range on its wake. Con¬ 
sequently no target strengths were measured at as¬ 
pect angles beyond about 110 degrees from the bow. 
Errors in the estimated aspect angles were quite 
large because of unknown deviations of the destroyer 
from its normal course but could not be evaluated. 
Ranges varied from 112 to 660 yd. 

Echoes from the destroyer and its wake were re¬ 
ceived and recorded oscillographically on moving 
picture film, with the equipment employed in the 


422 


TECHNIQUES OF NEW YORK MEASUREMENTS 


423 


reverberation studies. The average maximum ampli¬ 
tude of five successive echoes, together with the 
calibration constants of the equipment, was used to 
compute the echo level at each aspect angle; an 
auxiliary transducer measured the source level be¬ 
fore and after each run. However, the transmission 
loss was not measured directly. Transmission condi¬ 
tions were fair, since the water was isothermal to a 
depth of about 50 ft. Accordingly, inverse square 
divergence and an attenuation coefficient of 5 db per 
kyd were assumed in estimating the transmission 
loss. Each target strength was computed from the 
average echo level of five echoes; each range was the 
average range over the five echoes, as measured on 
the oscillograms. Aspect angles were estimated 
trigonometrically. 

24.2 TECHNIQUES OF NEW YORK 
MEASUREMENTS 

Two series of tests were made by BTL on ships in 
Long Island Sound early in 1944, as part of a specific 
development project to study the effects of short 
pulse lengths and receiver bandwidth on echo rang¬ 
ing, 3 and to measure echoes from surface vessels. 4 
Because little time was allocated to this part of the 
program, the work was discontinued as soon as 
enough data were obtained to establish the range of 
echo intensities to be expected. 

In the earlier measurements, made in Long Island 
Sound near City Island, Hart’s Island, and Execu¬ 
tion Light, pulse lengths from 0.05 to 150 msec were 
used at frequencies between 20 and 30 kc. 3 Echo¬ 
ranging gear including a transmitter and a receiving 
system of adjustable characteristics was mounted 
aboard a laboratory boat, the Elcobel, which was al¬ 
ready equipped with a standard Navy projector 
dome. Targets of these tests were various freighters 
in the vicinity. No absolute echo levels or target 
strengths were measured, since the experiment was 
conducted largely to investigate the effects of pulse 
and receiver characteristics on reverberation, noise, 
and echo character. Relative echo amplitudes were 
found for different pulse lengths, however, and are 
reported in Section 24.7. 

Later studies reported in more detail the reflect¬ 
ing characteristics of a total of twenty surface ves¬ 
sels. 4 In these measurements, a crystal transducer 
was mounted on the Elcobel, a 65-ft boat, in such a 
way that it could be carried just below the keel while 
under way, or lowered to a depth of 10 ft for echo 


ranging. In the lower position, the transducer could 
be trained by means of a hand wheel on top of the 
shaft; however, the speed of the Elcobel could not 
exceed a few knots without interfering with the satis¬ 
factory operation of the transducer. 

An oscillator aboard the Elcobel delivered pulses 
approximately 3 msec long to the transducer, at a 
frequency of 27 kc. The echoes received by the trans¬ 
ducer were amplified, observed and photographed on 
the screen of a cathode-ray oscilloscope, whose hori¬ 
zontal sweep was proportional to the time — and 
therefore to the range of the echo — and whose ver¬ 
tical sweep was proportional to the amplitude of the 
echo. 

In order to obtain target strengths, the average 
range and the average peak amplitude of between 10 
and 60 echoes were measured on the oscillograms. 
The transmission loss was estimated on the assump¬ 
tion of inverse square divergence and an attenuation 
coefficient of 7 db per kyd, although such an assump¬ 
tion probably was unrealistic since the water was 
shallow during these measurements and the ocean 
bottom was an effective reflector of sound. From the 
average range, the average peak amplitude, and the 
transmission loss, the diameter of the equivalent 
sphere was computed —- the sphere which woidd 
theoretically return the same echo under the same 
conditions. Then the target strength was readily 
determined from the diameter of the equivalent 
sphere by use of equation (10) in Chapter 19. 

24.2.1 Tests on Anchored Vessels 

In the first part of the second series of tests, the 
targets were ships at anchor in the tideway of Long 
Island Sound, near City Island, New York, where the 
water was less than 100 ft deep. Echoes from five 
freighters, a tanker, a Liberty ship, and a small 
British carrier were measured. During these tests the 
Elcobel was kept under way at a very slow speed, so 
that both the range and the aspect angle of the tar¬ 
get varied in almost all the tests. 

24.2.2 Tests on Moving Vessels 

The Elcobel also ranged on moving ships farther 
out in Long Island Sound, in the vicinity of Lloyd’s 
Neck, Long Island. These tests were made, without 
any advance arrangements, on passing ships whose 
courses brought them close enough to the Elcobel to 
make them satisfactory targets. Unfortunately, the 



424 


SURFACE VESSEL TARGET STRENGTHS 


speed of the Elcobrl had to be held down to a few 
knots when the transducer was in operation, while 
the target ships were traveling at least several times 
faster. Consequently a special procedure was de¬ 
veloped to fit these conditions. 

As the ship approached, the Elcobel maneuvered 
so that it neared the ship at an aspect angle just off 
the bow of the target ship. When the range was 
closed to about 600 yd, the test began, and the 
Elcobel followed a course that kept the transducer 
constantly aimed at the stern of the ship; the sound 
beam was wide enough to cover the entire ship even 
at close ranges. It is possible that echoes were also 
obtained from the wakes of the ships, although such 
echoes were probably distinguishable from ship 
echoes for pulse lengths of 3 msec. Observations were 
made as frequently as possible, and the range and 
aspect angle were estimated at the time of the ob¬ 
servations. Actually both ships deviated from their 
nominal courses because of the effects of the wind 
and sea state as well as inaccuracies in steering, so 
that the ranges and aspect angles changed rather 
irregularly. 


expected to depend on aspect angle in much the same 
way as the target strength of a submarine depends 
on its aspect. In fact, since most surface vessels are 
more nearly flat at beam aspects than submarines, a 
sharper dependence might be predicted as long as re¬ 
flections come exclusively from the hull, in other 
words, as long as the ship is anchored, or moving 
through the water very slowly, and gives rise only to 
specular reflection. If a moving ship reflects sound 
diffusely, some change of target strength with aspect 
angle might be expected, but not so marked a change 
close to beam aspect as results from specula re¬ 
flection. 

Table 1 lists beam and off-beam target strengths 
for still and moving ships, together with the ranges 
at which they were measured and the number of in¬ 
dividual observations — each comprising at least 
five echoes — which were averaged to obtain the 
tabulated results. Here the values given for beam 
target strength include all measurements at esti¬ 
mated aspect angles between 70 and 110, and be¬ 
tween 250 and 290 degrees from the bow, whereas 
off-beam target strengths include measurements at 


Table 1 Aspect dependence. 


Test 

Range 

in 

yards 

Beam 

target strengths 
and 

standard deviations 
in decibels 

Number 

of 

obser¬ 

vations 

Number 

of 

ships 

Range 

in 

yards 

Off-beam 
target strengths 
and 

standard deviations 
in decibels 

Number 

of 

obser¬ 

vations 

Number 

of 

ships 

Anchored ships 
(New York) 1 

250-587 

37.3 ± 16.3 

8 

5 

168-508 

13.3 ± 7.6 

23 

9 

Moving ships 
(San Diego) 1 

112-640 

20.8 ± 5.7 

42 

2 

140-660 

17.2 + 4.9 

35 

2 

Moving ships 
(New York) 1 

250-490 

16.2 ± 6.1 

62 

12 

300-562 

13.7 ± 5.1 

66 

12 


21.3 ASPECT DEPENDENCE 

Surface vessel target strengths have been measured 
at San Diego and New York for different aspect 
angles, ranges, speeds, types of ships, pulse lengths, 
and frequencies. A dependence on aspect angle and 
on range is suggested by the reported data; in addi¬ 
tion, the target strength of ships under way appears 
to be considerably different from the target strengths 
of still vessels. Sufficient information, however, is 
not available to permit evaluation of the effects of 
the class of ship, pidse length, or frequency on the 
target strengths measured. 

The target strength of a surface vessel might be 


all other aspect angles. Ranges, speeds, ship types, 
pulse lengths, and frequencies are not separated. 
Table 1 is illustrated graphically in Figure 1. 

24 . 3.1 Still Vessels 

Only for the anchored ships is the difference be¬ 
tween beam and off-beam target strengths roughly 
the same as the scatter of the observations, as repre¬ 
sented by the standard deviation. The dependence 
of target strength on aspect angle for the individual 
measurements on these still vessels is shown in Fig¬ 
ure 2. Two target strengths at an aspect angle of ap¬ 
proximately 100 degrees are conspicuously higher 

















ASPECT DEPENDENCE 


425 


BOW 



BEAM 


Curve Test 

-San Diego measurements on moving vessels. 

— — — New York measurements on anchored vessels. 

--— New York measurements on moving vessels. 

Figure 1. Aspect dependence for all tests. 

than values at any other aspect, almost 20 db higher 
than the next highest value, and more than 40 db 
higher than the average target strength at off-beam 
aspects. Such a peak would be expected theoreti¬ 
cally from specular reflection from the broadside of 
the ship at beam aspect; at a few degrees away from 
beam aspect, however, the target strength should be 
markedly reduced. 

This peak in Figure 2 may be exaggerated for two 
reasons, so that the actual aspect dependence may 
not be so sharp as it appears. First, the two observa¬ 
tions constituting the peak were made at ranges be¬ 
tween 500 and 600 yd, but most of the other observa¬ 
tions were made at much closer ranges. Since the sur- 



0 20 40 60 80 100 120 140 160 180 

BOW BEAM STERN 

ASPECT ANGLE IN OEGREES 


Figure 2. Aspect dependence for still vessels (New 
York). 

face vessel target strengths were also found to de¬ 
pend on the range, increasing as the range increased, 
especially at beam aspect, these high values may be 
more the result of the effect of the range on the tar¬ 
get strength, than the effect of the aspect angle of 
the target. The data were too few to permit separa¬ 
tion of these two factors and independent evaluation 
of the effect of each on the measured target strengths. 

Secondly, so few observations are plotted in Fig¬ 
ure 2 that, two very marked peaks may not be re¬ 
liable. Only 31 values were obtained in the New 
York tests on anchored vessels, and only 8 of these 
w r ere at aspects within 20 degrees of the beam. In 
view of the large scatter of values, the observations 
cannot be considered conclusive, but are at least gen¬ 
erally consistent with the theoretical expectation 
that still vessels reflect sound specularly. 


24.3.2 Moving Vessels 

The dependence of target strength on aspect angle 
for moving vessels is much less than the dependence 
found for still vessels. The small variation in Table 1 
is too small to be very significant. For comparison 
with Figure 2, the individual target strengths for 
moving ships measured at New York are plotted in 
Figure 3 and for the destroyers measured at San 
Diego in Figure 4. The scatter is so large that any 
possible systematic variation of target strength with 
aspect angle is largely obscured. 

Analysis of the San Diego data on moving vessels 
































426 


SURFACE VESSEL TARGET STRENGTHS 


0 

BOW 


20 



40 


60 


120 


140 


80 100 
BEAM 

ASPECT ANGLE IN DEGREES 

Figure 3. Aspect dependence for moving vessels (New York) 


160 180 

STERN 


<n 

Ui 

CO 

o 

UJ 

o 








• 

• 

. • 

• 

• 

• 

• • 

. 

.*• * 

• * 

TV- 

• 

• 

• 

• 

. 

, • t. 

• *. 

• • 

• • 

• 

• 





• 









0 20 40 60 80 100 120 

BOW BEAM 

ASPECT ANGLE IN DEGREES 


Figure 4. Aspect dependence for moving vessels (San 
Diego). 


leads to much the same results, which are illustrated 
in Figure 4. Since the target strengths measured at 
San Diego were found to depend markedly on range 
(see Section 24.4), they were separated according to 
range in order to examine the dependence on aspect 
angle. The data were broken down into three groups, 
for ranges of 100 to 300 yd, 300 to 500 yd, and 500 to 
700 yd, and each group was analyzed for a possible 
dependence on aspect angle. Average target strengths 
for aspects within 20 degrees of the beam, and for all 
other aspects are shown in Table 2 for each range 
group. 


Table 2. Aspect dependence at different ranges for 
moving destroyers (San Diego). 


Range 
in yards 

Beam 

target strength 
and 

standard deviation 
in decibels 

Off-beam 
target strength 
and 

standard deviation 
in decibels 

100 to 300 

12.3 ± 1.8 

13.0 ± 1.7 

300 to 500 

22.5 ± 3.6 

16.9 ± 4.8 

500 to 700 

23.9 ± 3.6 

20.9 ± 3.7 


The scatter of the individual values from the aver¬ 
ages in Table 2 is less than the scatter from the 
overall averages in Table 1, and a slight change of 
target strength with aspect seems significantly 
shown. Some dependence of this nature might be 
expected from a diffusely reflecting surface, if all the 
target were in the path of the sound beam. However, 
any change with aspect angle as great as that found 
for submerged submarines seems ruled out by Table 
2. These results are generally consistent with the 
hypothesis that bubbles along the side of the ship 
are responsible for the echoes observed from moving 
ships. More accurate data would be required, how¬ 
ever, for verification of this theory. 

24.4 R4NGE DEPENDENCE 

In Sections 20.4.4 and 23.4.1 it was pointed out 
that the target strength of submarine depends on 
the range at ranges less than the maximum radius 
































target strength in decibels . target strength in decibels 


RANGE DEPENDENCE 


427 



100 150 200 250 300 350 400 450 500 550 600 

RANGE IN YARDS 

Figure 5. Range dependence at beam aspects for anchored vessels (New York). 



Figure 6. Range dependence at off-beam aspects for anchored vessels (New York). 










































428 


SURFACE VESSEL TARGET STRENGTHS 



100 150 200 250 300 350 400 450 500 550 600 650 

RANGE IN YARDS 


Figure 7. Range dependence at beam aspects for moving vessels (San Diego). 



Figure 8. Range dependence at off-beam aspects for moving vessels (San Diego). 



RANGE IN YARDS 

Figure 9. Range dependence at beam aspects for 
moving vessels (New York). 

of curvature of the submarine. The target strength 
of a still ship would be expected to behave in the 
same way under similar conditions. Because ship 
hulls may be flatter and may have a larger radius of 


curvature than submarines, this dependence on 
range might extend to much longer ranges than for 
submarines. On the other hand, the target strength 
of a moving ship might be expected to increase as the 
range increases, as more and more of the scattering 
surface lies in the path of the direct sound beam. 

Accordingly, target strength was examined as a 
function of range, for beam and off-beam echoes, for 
all three sets of data. The results of this analysis are 
illustrated in Figures 5 to 10, where in each graph 
the solid line represents the least squares solution 
based on an assumed linear relation between the tar¬ 
get strength and the range. The slopes of these lines 
are listed in Table 3. It is apparent that in all cases 
the dependence of target strength on range is most 
pronounced (1) for still vessels and (2) at beam 
aspect. 

Three explanations may be suggested to account 
for the increase in target strength with range: (1) 
failure of the sound beam to cover the target at short 




















































RANGE DEPENDENCE 


429 



RANGE IN YARDS 


Figure 10. Range dependence at off-beam aspects for 
moving vessels (New York). 

ranges; (2) reduced reflection as the range approaches 
the dimensions of the target; and (3) incorrect evalu¬ 
ation of the transmission loss. The first effect applies 
only to the measurements on moving vessels at San 
Diego, since the sound beam used during the New 
York tests was wide enough to cover the target at all 
ranges. The second applies primarily to measure¬ 
ments on anchored vessels where specular reflection 
seems most likely to occur, and the third applies to 
measurements on both moving and still ships. 


Table 3. Range dependence. 


Test 

Slope of target strength-range curve 
in decibels per kiloyard 


Beam aspect 

Off-beam aspects 

Anchored ships 
(New York) 

105.0 

68.0 

Moving ships 
(San Diego) 

30.5 

24.4 

Moving ships 
(New York) 

26.3 

4.5 


At short ranges, how much of the target the sound 
beam covers depends on the dimensions and aspect 
angle of the target and on the directivity pattern of 
the transducer. At San Diego, a standard JIv trans¬ 
ducer was employed, which had a total beam width 
of 20 degrees between points on either side of the 
axis where the response was 10 db lower. II it is 
assumed that the sound beam was 20 degrees wide 
and that the destroyer was 300 ft long, then the 
sound beam did not cover the ship, at beam aspect, 
at ranges less than about 300 yd. Since many of the 
beam target strengths were measured at shorter 
ranges, this failure of the sound beam to cover the 
ship may account for the decrease in target strength 
with decreasing range. 


24.4.1 Transducer Directivity 

To evaluate the effect of the transducer directivity 
on the target strength-range dependence, the differ¬ 
ence between the echo level from a destroyer at beam 
aspect and from a small target always within the 
sound beam was calculated, as a function of range, 
from the directivity pattern of the transducer. This 
difference is expressed as 

, r n b-(<t>)dx X\ 

101 °g I TV . 8 . 8 - ~ 10 log (1) 

Jo ( x- + r 1 ) 1 r 4 

where x x is the length of the target in a direction per¬ 
pendicular to the sound beam; b 2 (<t >) is the composite 
directivity pattern of the transducer; and r the range 
to the center of the target. This difference in decibels 
between the echo level from the destroyer and the 
echo level from a small target of the same target 
strength, as the range is decreased from 050 to 100 
yd, is superimposed on Figure 7 as a broken line. The 
zero level, where the sound beam effectively covers 
the entire target and the two echo levels are the 
same, is placed at a target strength of 23.5 db, which 
is the average beam target strength measured at San 
Diego at ranges of 450 yd and greater. The difference 
calculated from equation (1) amounts to about 7 db 
at a range of 100 yd, and drops to less than 1 db at 
ranges greater than 500 yd. 

This analysis does not take into account the ex¬ 
tension of the target by the wake. However, even if 
the target were assumed to extend infinitely in one 
direction, the target strength for long pulses would 
increase only as 10 log r and would not be signifi¬ 
cantly different from the broken curve in Figure 7. 
The increase of target strength with range in such a 
case would be analogous to the similar increase for 
the target strength of wakes discussed in Section 
33.1.1. This failure of the sound beam to cover the en¬ 
tire target, especially at short ranges, is responsible for 
much of the dependence of target strength on range 
observed at San Diego at beam aspect. Apparently, 
however, it is not responsible for all the dependence 
observed. 

Significantly, these echoes from destroyers at beam 
aspect are approximately as strong as echoes ob¬ 
served from the wakes directly behind the destroy¬ 
ers. In Section 24.1 it was noted that echo-ranging 
experiments were made on the wakes after the de¬ 
stroyers passed; in these measurements, the sound 
beam was perpendicular to the axis of the wake. 
The wake echoes showed the same variation with 




























430 


SURFACE VESSEL TARGET STRENGTHS 


range as the destroyer echoes, and, like them, showed 
no significant dependence on speed. Differences as 
great as 10 db were observed between the wake 
echoes and the destroyer echoes immediately pre¬ 
ceding them, but these differences appeared to be 
quite random and unsystematic. This equivalency 
between wake echoes and destroyer echoes is con¬ 
sistent with the theory that both arise from scatter¬ 
ing by small bubbles. 

The dependence of target strength on range at off- 
beam aspects from the San Diego results shown in 
Figure 8 cannot be analyzed very simply. In the first 
place, the spread of aspect angles covered is very 
wide; the projected length of the destroyer, measured 
in a direction perpendicular to the sound beam, 
varied from 30 ft at bow and stern aspects to 290 ft 
at an aspect angle 20 degrees from the beam. In the 
second place, with a pulse length of 10 msec, the en¬ 
tire destroyer was not in the sound beam at the same 
time, especially at bow and stern aspects; for aspects 
close to the beam, this effect of pulse length may be 
neglected, but it becomes important at other aspects. 
When more of the target comes into the sound beam, 
the observed echo from a very short pulse will not be 
stronger but instead will last longer, as pointed out 
in Section 19.3. Thus the change of target strength 
with range at off-beam aspects cannot be explained 
even in part by this simple mechanism. 

Transducer directivity is also relatively unim¬ 
portant in the New York measurements on still and 
moving vessels since a very wide beam was employed. 
The total horizontal beam width of the combined 
projector-hydrophone directivity pattern, between 
points where the response was 10 db lower than on 
its axis, was about 40 degrees, which even at a range 
of 168 yd, the shortest range at which measurements 
during either test were made, still covers the long¬ 
est ship at beam aspect. Therefore the decrease in 
target strength with decreasing range in the New 
York measurements cannot be explained as a result 
of the failure of the sound beam to cover the target. 

24.4.2 Predicted Dependence 

The second explanation which might be suggested 
for the observed dependence of target strength on 
range is the predicted decrease of specular reflection 
with decreasing range for ranges less than the maxi¬ 
mum radius of curvature of the target, providing the 
reflection is specular (see Section 20.4.4). This 
effect would apply only to echoes from vessels sta¬ 


tionary in the water, which presumably arise prima¬ 
rily from the hull and not from a uniformly scatter¬ 
ing layer. However, in the most extreme case, reflec¬ 
tion from an infinite plane surface, the target strength 
will not vary more rapidly than as the square of the 
range. Such a variation is quite insufficient to ac¬ 
count for the large effect observed during echo¬ 
ranging trials on still vessels illustrated in Figures 5 
and 6. Qualitatively, however, it partly explains the 
difference in range dependence at beam and off-beam 
aspects, since the radius of curvature of the ship is 
greater when it presents its broadside to the incident 
sound than when it is at bow or stern aspect. 

24.4.3 Transmission Loss 

A third possible explanation of the observed range 
dependence is a possible incorrect evaluation of the 
transmission loss. In none of the measurements was 
the transmission loss measured directly. Instead, an 
attempt was made to estimate it from the prevailing 
conditions on the basis of inverse-square divergence 
and an additional attenuation proportional to the 
range. 

At San Diego, it was assumed that the intensity of 
the echo was inversely proportional to the fourth 
power of the range, weakened by an additional loss 
of 5 db per kyd of sound travel. The water was 
isothermal to a depth of 50 ft, so that an assumption 
of 5 db per kyd for the attenuation coefficient seems 
somewhat low. Use of equation (1) in Chapter 23 
gives an attenuation coefficient of about 7 db per 
kyd. An attenuation coefficient of about 10 db per 
kyd would be required to explain the departure of 
the plotted points from the theoretical curve in Fig¬ 
ure 7. Such a high coefficient does not seem very 
likely when the surface layer is isothermal down to 
a depth of 50 ft, but it is not impossible. 

At New York, the transmission loss was assumed 
to follow the same inverse square loss with an atten¬ 
uation coefficient at 27 kc of 7 db per kyd. The tem¬ 
perature conditions of the water were not known; 
the wind velocity varied from 1 to 23 mph. Whether 
or not the assumed attenuation coefficient is reliable 
it is difficult to say. In addition, bottom-reflected 
sound may have had a marked effect on the trans¬ 
mission loss. 

Conditions were very favorable to bottom reflec¬ 
tion during these New York tests. The bottom was 
composed of sand and mud, a mixture which reflects 
sound very effectively. In addition, the water was 



DEPENDENCE ON SHIP TYPE 


431 


very shallow, from 60 to 110 ft deep. The sound beam 
was not highly directional; the total vertical beam 
width between points where the response was 10 db 
down was 20 degrees. Thus if the transducer were 
level, the sound beam would strike the bottom at a 
range of only about 126 yd, for water 60 ft deep. 
Consequently the bottom undoubtedly reflected part 
of the incident sound in much the same way as the 
surface, and contributed to the intensity of the 
echoes received at the transducer. 

Assume, for example, that both the surface and 
bottom reflected sound perfectly, so that at the 
particular ranges used the sound beam could spread 
in only one direction — horizontally. In this extreme 
case, the intensity of the echo would be inversely pro¬ 
portional, not to the fourth power of the range, but 
to the square of the range. This assumption,of course, 
is not realistic, but the result suggests that for the 
New York measurements the actual drop is some¬ 
where between inverse fourth and inverse square; 
perhaps the echo intensity actually varies more 
nearly inversely as the cube of the range over a shal¬ 
low reflecting bottom. This relatively slow increase 
of transmission loss with increasing range may ac¬ 
count for much of the range dependence for moving 
vessels in the New York data. Even an inverse 
square dependence of echo level on range fails to 
account, however, for the observed variation on still 
vessels shown in Figures 5 and 6, where the echo 
level actually increases rapidly with increasing 
range. 

Another possibility might account for the depend¬ 
ence of target strength on range for stationary ves¬ 
sels. Sound incident on the hull of the ship will be 
reflected downward where the hull is curved slightly 
downward, then reflected upward from the bottom. 
It is possible that the curvature of the hulls of the 
surface vessels measured is such that the rays re¬ 
flected to the bottom will strike the bottom and be 
reflected back to the transducer only at longer 
ranges, so that target strengths measured at long 
ranges will be greater than target strengths measured 
at short ranges. This explanation may account for 
the stronger range dependence for stationary ves¬ 
sels than for moving vessels, although it must be re¬ 
garded as highly tentative in the absence of further 
substantiating evidence. 

In all, it has been well established that the target 
strength of surface vessels on which measurements 
have been made apparently increases with range. 
This increase is much greater at beam aspects than 


at off-beam aspects. The exact rate of increase is un¬ 
certain because many causes are responsible; meas¬ 
ured rates vary from 4.5 to 105 db per kyd at ranges 
between 200 and 500 yd. The dependence of target 
strength on range arises from (1) smaller coverage of 
the target by directive transducers at close ranges; 
(2) incorrect evaluation of the transmission loss 
neglecting surface and bottom reflections; and (3) 
the dimensions and curvature of the target, in so far 
as they reduce specular reflection at close ranges. 
Probably none of these effects, however, can explain 
the enormous observed range dependence for an¬ 
chored vessels. Further measurements would be re¬ 
quired to show the extent to which this observed 
effect is generally found. 

24.5 DEPENDENCE ON SPEED 

Very little information is available on the varia¬ 
tion of target strength with the speed of the ship, for 
moving vessels. At San Diego, speeds of 10, 15, and 
20 knots were employed; Table 1 lists target strengths 
without separating the speeds at which they were 



RANGE IN YARDS 


Figure 11. Range dependence at different speeds for 
beam aspect (San Diego). 

measured. Figure 11 shows beam target strengths 
plotted as a function of range for three different 
speeds, 10, 15, and 20 knots, from the San Diego 
measurements. The dependence on range is evident, 
even in only twenty observations, but no significant 
dependence on ship speed is apparent. Ship speeds 
were not estimated or measured in the New York 
tests. As already mentioned, this same lack of de¬ 
pendence on ship speed is characteristic of wake 
echoes. 

24.6 DEPENDENCE ON SHIP TYPE 

No clear dependence of target strength on ship 
type is indicated by the evidence now available. 
While a large number of different vessels have been 



















432 


SURFACE VESSEL TARGET STRENGTHS 



ASPECT ANGLE IN DEGREES 




Length 

Draft 

Water 

Range 

Ship 

Tonnage 

in 

in 

depth 

in 



feet 

feet 

in feet 

yards 

Navy Transport 

12,000 

325 

25 

100 

300-510 

Liberty Ship 

10,000 

300 

20 

60 

300-450 


Figure 12. Variation in target strength between simi¬ 
lar ships (New York). 


their estimated tonnages, lengths, and drafts; fur¬ 
thermore, both were measured at roughly the same 
ranges. It is possible that the fluctuation and varia¬ 
tion normally encountered in underwater sound 
transmission may be responsible for the 10 db differ¬ 
ence between the two curves, or that bottom reflec¬ 
tion may be the cause, since the transport was under 
way in water 100 ft deep while the Liberty ship was 
under way in water almost half as deep. Even per¬ 
fect bottom reflection, however, cannot account for 
the observed difference between the two curves, 
which suggests faulty calibration, widely variable 
transmission, or large unsuspected systematic differ¬ 
ences between the two ships. 



o.oi 0.1 1.0 10 100 1000 

PULSE LENGTH IN MILLISECONDS 


Curve 


Range in yards 

Approximate aspect 
angle in degrees 

— 

180 

700 

45 

1600 

— 

2200 

180 


Figure 13. Effect of pulse length on measured echo levels (New York). 


made, 4 the scatter is so great that any correlation be¬ 
tween target strength and ship draft and tonnage is 
obscured. 

As an example of the variation in target strength 
between one ship and another, as measured at New 
York, Figure 12 illustrates target strength plotted 
against aspect angle for two large ships of nearly 
equal dimensions. The difference in their target 
strengths cannot be attributed to the difference in 


24.7 DEPENDENCE ON PULSE LENGTH 
AND FREQUENCY 

Although surface vessel target strengths have not 
been systematically investigated as a function of 
pulse length, early studies at New York reported a 
dependence of echo amplitude on pulse length for 
pulse lengths of 0.05 to 110 msec. 3 The results of 
these measurements are reproduced in Figure 13, 











































































DEPENDENCE ON PULSE LENGTH AND FREQUENCY 


433 


where the relative echo level in decibels is plotted 
against the pulse length for four freighters. Little 
dependence on pulse length is evident for pulses more 
than 10 msec long, in qualitative agreement with 
the results described in Section 23.5.2 applying to 
submerged submarines. However, for pulse lengths of 
less than 10 msec, the echo level drops rather sharply. 
More data are required, however, to show how great 
this dependence will be for any actual vessel. 

No information is available on how surface vessel 
target strengths vary with the frequency of the echo¬ 
ranging beam employed. The only tests were made 


at San Diego at 24 kc and at New York at 27 kc; any 
difference in the target strengths at these two fre¬ 
quencies would probably be very small, from theo¬ 
retical predictions, and the actual measured differ¬ 
ence is too small to verify any such dependence. For 
still vessels, if the echo comes from the hull, very 
little variation of target strength with frequency 
would be expected (see Sections 20.2 and 20.3). For 
moving vessels, however, with sound scattered from 
a layer of bubbles, the target strength would be ex¬ 
pected to vary with frequency in accordance with 
the acoustic properties of small bubbles. 



Chapter 25 


SUMMARY 


25.1 DEFINITION OF TARGET STRENGTH 25.1.3 Transmission Loss 


F or the purposes of discussing the reflecting 
characteristics of different vessels, the target 
strength T of a target is defined by 

T = E - S + 2H, (1) 

where E is the echo level, S the source level, and H 
the one-way transmission loss from the source to the 
target, all in decibels (see Section 19.1.3). For most 
targets, T is independent of range at ranges much 
greater than the dimensions of the target (see Sec¬ 
tions 20.4 and 23.4), but may change with the chang¬ 
ing orientation of the target relative to the sound 
beam (see Section 23.1). 

25.1.1 Echo Level 

The echo level E is defined by 

E = 20 log p e , (2) 

where p, is the rms pressure of the echo, in dynes per 
square centimeter averaged over a few cycles (see Sec¬ 
tion 19.1.3). If the rms pressure is not constant dur¬ 
ing the echo, E is defined as the peak rms pressure. 

25.1.2 Source Level 

For directional supersonic projectors, the rms pres¬ 
sure p of the sound on the axis of a projector is in¬ 
versely proportional to the square of the range r, as 
long as the range is much greater than the dimen¬ 
sions of the target and as long as the range is small 
enough so that attenuation and surface reflection 
may be neglected. Under these conditions, the source 
level S is defined by 

S = 20 log p + 20 log r, (3) 

where p is the pressure of the sound on the axis of 
the projector, in dynes per square centimeter, at a 
distance r, in yards, from the projector (see Section 
19.1.3). 


The difference between the pressure level of the 
transmitted sound at some point, and the source level 
is called the transmission loss // from the projector 
to that point (see Section 19.1.2). 

25.1.4 Average Values 

Since both E and H often fluctuate by as much as 
10 db from pulse to pulse, it is customary to use the 
average echo amplitude in determining E, and the 
average pressure amplitude at the range r in deter¬ 
mining //, in equation (1), where the average ampli¬ 
tude is the average of a number of peak rms ampli¬ 
tudes, if the rms amplitude is not constant over each 
echo (see Section 21.G.4). 


25.1.5 Target Strength of Sphere 

A sphere reflects a plane wave equally in all direc¬ 
tions (see Section 19.2.2). The target strength of a 
sphere is 

T = 20 log | (4) 

where A is the radius of the sphere in yards (see Sec¬ 
tions 19.2.1, 19.2.2 and 20.4.1). This formula is accu¬ 
rate to 0.5 db if the range to the sphere is greater 
than ten times its radius, and if the wavelength of 
the sound is less than the radius of the sphere. 


25.1.6 Target Strength of a General 
Convex Surface 


The target strength for specular reflection from 
any convex surface is 


10 log 


A x A 2 


4 1+- 


tx 


a 2 
i + — 

r 


(5) 


where A\ and A 2 are the principal radii of curvature 
of the target surface at the point where the surface is 


434 



BEAM ECHOES FROM SUBMERGED SUBMARINES 


435 


perpendicular to the sound beam, and r is the range 
(see Section 20.4.2). This formula is valid only if 
both Ai and A 2 are greater than the wavelength of 
the sound, and if either Ai or A 2 is much less than r. 

25.1.7 Target Strength of a Cylinder 

For a cylinder, A 2 is infinite in equation (5) and the 
target strength becomes 



This equation is valid only when the cylinder radius 
Ai is less than r and the wavelength is less than A u 

25.2 BEAM ECHOES FROM SUBMERGED 
SUBMARINES 

At aspect angles within about 20 degrees of the 
beam, echoes from submarines are produced prima¬ 
rily by specular reflection from the pressure hull, the 
fuel and ballast tanks, and the conning tower (see 
Section 23.8.1) and are much stronger than echoes 
at other aspects. Oscillograms show that the echo 
generally reproduces the outgoing pulse (see Figures 
25 and 27 in Chapter 23). 

25.2.1 Beam Target Strengths 

Observed submarine target strengths at beam as¬ 
pects and at long ranges lie mostly between 20 and 
30 db. About 25 db is the average value (see Section 
23.1.1); typical values of Ai or A 2 in equation (4) 
which would correspond to this target strength would 
be 500 and 2.5 yd respectively. The observed spread 
of values may result entirely from experimental 
errors. 

Off-beam target strengths, found at aspects 20 
degrees or more away from the beam, are reported 
in Section 25.3. 

Variation with Submarine Class 
Observed differences in the target strengths of 
different submarines measured both directly and in¬ 
directly are less than the estimated experimental 
error in the direct measurements (see Section 25.2.1). 
Consequently no reliable overall evaluation of the 
dependence of the target strength on the class of sub¬ 
marine can be made. 

Variation with Submarine Speed 

No significant variation of target strength with 
submarine speed is expected, since the wake of a sub¬ 


merged submarine is a poor reflector of sound (see 
Section 33.3). No pronounced variation has been ob¬ 
served in practice for submerged speeds from 1 to 6 
knots at keel depths of about 100 ft (see Section 23.3). 

Variation with Range 

Theoretically, beam target strengths depend on 
the range at ranges less than the principal radii of 
curvature of the submarine at beam aspect (see Sec¬ 
tions 20.4.4 and 23.4). For a 517-ton German U- 
boat, approximated by an ellipsoid with principal 
radii of curvature of 576 and 2.3 yd, the variation of 
target with range predicted from equation (5) is 
shown in Table 1. Although no observations are 


Table 1. Theoretical range variation. 


Range 

in 

yards 

Submarine 
target strength 
beam aspect 
(without conning 
tower) 

Submarine 
target strength 
beam aspect 
(with conning tower) 

8 

5.5 

5.8 

12 

7.5 

7.4 

16 

8.9 

9.1 

200 

19.2 

22.9 

1,000 

23.2 

25.5 

CO 

25.2 



available to confirm this variation with range, the 
result is believed to be reliable. 

Variation with Pulse Length 
No marked dependence of target strength on pulse 
length is expected at beam aspect, since the echo 
approximately reproduces the pulse (see Section 
23.5.1). The available evidence is neither very con¬ 
sistent nor conclusive, but does not demonstrate any 
sharp variation in the target strength with the pulse 
length (see Section 23.5.2). 

Variation with Frequency 

No variation of target strength with frequency is 
expected theoretically at beam aspects for specular 
reflection (see Sections 20.4 and 23.6.1). Observa¬ 
tions confirm this prediction (see Section 23.6.2), ex¬ 
cept for a few measurements at 60 kc; these 60-kc 
target strengths, however, are so large that calibra¬ 
tion errors are believed responsible. 

25.2.2 Echo Structure 

Generally, beam echoes are square-topped and 
resemble the outgoing pulses (see Section 23.8.1). 
For very short pulses, beam echoes from submarines 











SUMMARY 


436 


reveal a definite structure. For observations on one 
S-boat, the main echo consists of two components 
separated by a distance of about 4 yd; the first com¬ 
ponent may come from the broadside of the sub¬ 
marine, while the second component may be an echo 
from the bilge keel or conning tower. After this main 
echo comes a much weaker secondary echo, pre¬ 
sumably resulting from sound reflected from the 
submarine straight up to the surface, back down to 
the submarine, and then back to the projector (see 
Figure 9 in Chapter 21. and Figure 29 in Chapter 23). 
The presence of this echo structure will be expected 
to modify slightly the conclusions in the preceding 
section, since for long pulses the different components 
will combine. Such a combination will increase the 
average target strength 3 db at most above its value 
for very short pulses, 

25.2.3 Fluctuation 

The fluctuation of beam echoes may be primarily 
attributed to the fluctuation in the transmission of 
the outgoing and incoming sound (see Section 21.6). 
Much of this fluctuation is apparently due to the 
presence of surface-reflected sound (see Section 
21.5.4). Estimates of the fluctuation of transmitted 
sound are given in Chapters 7 and 10. In addition, 
for pulses more than a few milliseconds long, inter¬ 
ference between the different components of the echo 
will somewhat increase the fluctuation. 

25.3 OFF-BEAM ECHOES FROM SUB¬ 

MERGED SUBMARINES 

At aspect angles more than about 20 degrees off 
the beam, echoes from submarines originate along 
the entire length of the vessel and probably result 
from both specular and nonspecular reflection (see 
Section 23.8.2); they are 10 to 15 db weaker than 
echoes at beam aspect. The echo does not reproduce 
the outgoing pulse (see Figures 25 and 28 in Chapter 
23). 

25.3.1 Off-Beam Target Strengths 

Observed submarine target strengths at off-beam 
aspects and at long ranges lie mostly between 5 and 
20 db for pulses 100 or more msec long, and usually 
between 10 and 15 db (see Section 23.1.1). The 
spread of values is apparently real to some extent, 
since at different aspect angles echo characteristics 
are markedly different. At certain off-beam aspects 


and altitudes, strong specular reflections from nearly 
flat surfaces, such as the conning tower, may give 
target strengths greater than 20 db (see Section 
23.2.2); these reflections depend critically on the 
particular submarine measured. 

Variation with Submarine Class 

No variation in the off-beam target strengths of 
different submarines has been observed in either the 
direct or indirect measurements to be greater than 
the estimated experimental error in the direct meas¬ 
urements (see Section 25.2.1). 

Variation with Submarine Speed 

No important variation of target strength with 
submarine speed has been observed at off-beam as¬ 
pects (see Section 25.2.1). 

Variation with Range 

At off-beam aspects, submarine target strengths 
decrease with decreasing range (see Section 25.2.1). 
At ranges less than the length of the submarine, 
off-beam target strengths are roughly equal to beam 
target strengths. Under such conditions, a submarine 
may be approximated by a cylinder at off-beam as¬ 
pects except bow and stern, and equation (6) may 
be used. 

Variation with Pulse Length 
Since at off-beam aspects the echo does not usually 
reproduce the pulse and the echo length considerably 
exceeds the pulse length, for pulses 100 or more msec 
long, some variation of target strength with pulse 
length may be expected (see Section 23.5.1). Ob¬ 
served target strengths decrease with pulse length 
for signals shorter than 100 msec (see Section 23.5.2). 
The decrease is most marked for pulses shorter than 
10 msec, but even for such short pulses the target 
strength does not decrease as rapidly as the pulse 
length, or rather, as rapidly as 10 log r, where r is the 
pulse length in milliseconds. 

Variation with Frequency 

No variation of target strength with frequency is 
expected at off-beam aspects (see Section 23.6.1). 
This conclusion is contradicted by some target 
strength measurements at a frequency of 60 kc, which 
give much higher results than similar measurements 
at 24 kc (see Section 25.2.1). However, the differences 
between beam and off-beam target strengths are 
about the same at 60 kc as at 24 kc, so that if the ob¬ 
served frequency effect is real, it is the same at all 
aspects. 



ECHOES FROM SURFACE VESSELS 


437 


25.3.2 Echo Structure 

At off-beam aspects, echoes from submarines do 
not reproduce the outgoing pulses because the entire 
length of the submarine reflects sounds (see Section 
23.8.2). The duration of the echo, measured on an 
oscillogram, may be given by 

m 2 L 

T = — cos 6 -T t, (7) 

c 

where T is the duration of the echo, L the length of 
the submarine, c the velocity of sound, 6 the aspect 
angle measured from the bow of the submarine, and 
t the pulse length. On a sound range recorder, how¬ 
ever, the echo length is about half that given in 
equation (7), perhaps because only the stronger part 
of the echo woidd be expected to show on a recorder 
using chemically treated paper (see Figure 25 in 
Chapter 23). 

25.3.3 Fluctuation 

The fluctuation of echoes at off-beam aspects is 
due not only to fluctuations in the transmission of 
the sound each way (see Section 25.2.3), but also to 
fluctuations resulting from interference phenomena. 
The echo obtained from a long pulse will be the re¬ 
sult of constructive and destructive interference be¬ 
tween echoes from individual reflecting surfaces dis¬ 
tributed over the length of the submarine. Changes 
in this interference pattern as the aspect or altitude 
of the submarine changes slightly will increase the 
observed fluctuation of echoes. 

25.4 ECHOES FROM SURFACE VESSELS 

Information on reflection from surface vessels is 
even more fragmentary than on reflection from sub¬ 
marines. The following conclusions are suggested by 
the data but cannot all be regarded as confirmed. 

25.4.1 Still Vessels 

Vessels at anchor seem to behave as targets in the 
same way as submerged submarines. At aspects close 
to the beam, target strengths may be very high, as 


much as 40 db, but at other aspects, for pulses 3 msec 
long at a frequency of 27 kc, it is usually between 5 
and 20 db (see Section 24.3.1). The strong echoes at 
beam aspect are presumably the result of specular re¬ 
flection from the hull of the ship. 

25.4.2 Moving Vessels 

When a vessel is under way, beam echoes are about 
the same as off-beam echoes (see Section 24.3.2). Ob¬ 
served target strengths of moving destroyers and 
merchant vessels lie between 10 and 25 db, for pulses 
3 and 10 msec long at frequencies of 24 and 27 kc; a 
systematic difference in the target strengths of differ¬ 
ent ships is not evident (see Section 24.6). An in¬ 
crease in speed from 10 to 20 knots apparently does 
not affect the target strength appreciably (see Sec¬ 
tion 24.5). A decrease in pulse length decreases the 
resultant target strength, especially for pulse lengths 
less than 10 msec, but the target strength does not 
drop as rapidly as 10 log r, where r is the pulse length 
(see Section 24.7). 

Echoes from moving vessels may arise from scat¬ 
tering by bubbles of entrained air along the side of 
the ship. This implies that the echo from a moving 
ship may be treated as an echo from a short stretch 
of wake (see Section 33.1.1). 

25.4.3 Dependence on Range 

Most of the data on target strengths of moving 
vessels show a marked increase in target strength as 
the range increases from 200 to 600 yd, in one case 
amounting to more than 30 db (see Section 24.4). 
Although some increase is expected from the geom¬ 
etry of the ship (see Sections 24.4.2 and 25.2.1) and 
from the failure of the sound beam to cover the en¬ 
tire ship at short ranges (see Section 24.4.1), so 
marked a change seems greater than can be explained 
on any simple basis; it is quite possibly a statistical 
accident. Beyond about 600 yd, it is reasonable to 
assume that the target strength does not depend on 
the range, and that its value lies within the spread 
specified for surface vessel target strengths at off- 
beam aspects in the preceding section. 




















































PART IV 


ACOUSTIC PROPERTIES OF WAKES 























Chapter 26 


INTRODUCTION 


26.1 WHAT 4RE WAKES? 

he appearance of a streak of foamy, churned 
water behind a ship under way, known as the 
ship’s wake, is familiar to every mariner. Because the 
wake extends along the path of the ship over a 
length many times the ship’s length, it is hard to get 
a good view of the wake as a whole, even if a some- 
what elevated vantage point, such as the bridge or 
masthead, is chosen. Figure 1 shows the wake of an 
antisubmarine patrol vessel (PC488) in a quiet sea, 
as viewed aft from the crow’s nest. 

The observer in an airplane enjoys ideal condi¬ 
tions for the visual study of wakes. Figures 2 to 6 
describe better than verbal descriptions what a wake 
looks like from a great height. The first four were 
taken from altitudes of 2,500 to 3,000 ft, the plane in 
level flight overtaking a destroyer, the USS Moale 
(DD693), which was proceeding on a straight course 
at constant speeds of 16, 20, 25, and 33 knots. By 
way of a scale, the ship had an overall length of 
376 ft, a beam of 41 ft, and a draft of 13 ft. Figure 6 
illustrates what happens to the wake as the ship 
turns; the foam on the curved section of the path is 
seen to be visually more dense, especially along the 
outer edge of the wake. As in turning, acceleration of 
the ship on a straight course increases the visual den¬ 
sity of a wake. Incidentally, the irregular white 
streaks appearing in Figures 4 to 6 are foam patterns 
on the water. All the photographs show the delicate 
bow-wave pattern, fanning out astern with a much 
greater angle of divergence than the actual wake. 
Figure 7 is a close-up taken from an altitude of 300 
ft, of the bow wave and the wake of another de¬ 
stroyer, the USS Ringgold (DD500). The visible 
structure of the wakes laid by large ships does not 
differ markedly from that of the wakes of vessels of 
destroyer size, as may i>e seen in Figure 8, which 
gives a view from an altitude of 4,000 ft of a large 
aircraft carrier, the USS Saratoga (CV3). 

Beyond the obviously foamy and turbulent nature 


of wakes, visual inspection does not reveal any of 
their physical properties. The discovery that 
“wakes,” using this term in a loose sense, are capable 
of affecting the propagation of sound energy through 
the water has suggested a new distinction: an acoustic 
wake is defined as a volume of the ocean which has 
acquired, because of the passage of a ship through it, 
a greater though transitory capacity for absorbing 
and scattering sound. The expression,“volume of the 
ocean” is used advisedly, because acoustic wakes 
have a definite vertical extension, often rather 
sharply bounded. Acoustic wakes under the surface, 
originating from submerged submarines, are of par¬ 
ticular interest. During a level run at periscope 
depth, the upper boundary of the wake, spreading out 
bodily from the screws, does not reach the ocean sur¬ 
face until several hundred yards behind the sub¬ 
marine. Several aerial views of submarine wakes, 
both during surface runs and after a crash dive, are 
reproduced in Figures 1 to 6 of Chapter 31. 

The temperature distribution in the top layer of 
the ocean may be disturbed by the passage of a ship 
in such a manner as to leave a thermal wake, detect¬ 
able by sensitive thermocouples. Evidently, experi¬ 
ments must decide to what extent thermal and 
acoustic wakes coincide with the body of water called 
a wake by a visual observer. This problem is dis¬ 
cussed in Chapter 31, dealing with the geometry of 
wakes. However, one interesting feature will be men¬ 
tioned here: the acoustic and thermal manifestations 
of a wake may persist over periods of half an hour or 
more, often long after visible traces of the ship’s 
passage have disappeared. 

26.2 NAVAL IMPORTANCE OF WAKES 

Wakes can be important in naval warfare in two 
general ways. In the first place, they may interfere 
with the successful operation of acoustic devices, by 
scattering or absorbing sound. In the second place, 
they may provide a method for detecting, tracking, 



441 


442 


INTRODUCTION 



* i 


Figure 1 . Wake of submarine chaser (PC 488), seen from crow’s nest. 


jggggj 

£Ppjs|li 


§i®i|g 



















NAVAL IMPORTANCE OF WAKES 


443 



Figure 2. Wake of USS Moale (DD 693) at 16 knots from 2,500 feet. 


or identifying the ship which has produced the wake. 
Such utilization of wakes in offensive operations com¬ 
prises visual detection from the air and thermal de¬ 
tection from surface ships or submarines, as well as 
acoustic detection. However, the present discussion 
is concerned only with the acoustic properties of 
wakes and the importance of these acoustic proper¬ 
ties in naval warfare. 

Acoustic interference produced by wakes is fre¬ 
quently encountered in the operation of sonar gear. 
False echoes from submarine wakes may confuse the 
sonar operator on an antisubmarine vessel and may 
even lead to an attack on a wake knuckle, a dis¬ 
turbance in the water when a submarine suddenly 
speeds up and turns sharply, while the submarine 
escapes. During thirty unsuccessful attacks on sub¬ 
marines by United States antisubmarine vessels in 
1944, where the presence of a submarine was ascer¬ 
tained but no damage inflicted, 12 per cent of the 
failures were attributed to attacking wakes, a larger 
percentage than assigned to any other single cause. 

Wakes laid by surface vessels can also be disturb¬ 
ing in antisubmarine warfare. After one or more at¬ 
tacks in an area, echoes from old wakes from surface 


ships can be confusing. Moreover, a moderately fresh 
wake is highly absorbent and may shield a shallow 
target on one side of the wake from detection by a 
surface vessel on the other side. In fact a projector 
surrounded by a fresh wake is almost completely 
useless, since very little sound can escape through the 
w'ake. Thus a surface ship will commonly find that 
its echo-ranging equipment “goes dead” when the 
ship passes through a fresh wake. 

Harbor detection equipment can also be seriously 
hampered by the presence of wakes. When a de¬ 
stroyer at moderate speed passes in the neighborhood 
of bottom-mounted supersonic listening gear, ships 
passing by subsequently cannot be heard for some 
time. Similarly, sneak craft in the wake of a large sur¬ 
face vessel are very difficult to detect by echo rang¬ 
ing. To reduce the seriousness of these effects in 
combating submarines, or to use them most effec¬ 
tively in submarine warfare, accurate information is 
required on the reflection and absorption of sound by 
wakes under different conditions. 

The use of wakes in offensive operations against 
the wake-laying vessel is a relatively new field. As an 
example of this utilization of wakes, it was at one 





444 


INTRODUCTION 



Figure 3. Wake of USS Moale (DD 693) at 20 knots from 2,500 feet 












naval importance of m akes 


445 



Figure 4. Make of USS Moale (DD 693) at 25 knots from 3,030 feet. 



Figure 5. Wake of USS Moale (DD 693) at 33 knots from 2,500 feet. 














446 


INTRODUCTION 



Figure 6. Wake of USS Moale (DD 693) as ship turns at 30 knots. 


time suggested that attacks on submarines could be 
made by detecting the wake and then following it 
until the submarine was reached. This suggested pro¬ 
cedure turned out to be impractical, owing to the 
very low scattering power of the wakes behind slow, 
deep submarines. The wake laid by a surface vessel 
reflects sound so strongly and so persistently that 
acoustic methods might possibly be useful for at¬ 
tacks on such enemy vessels. Obviously a knowledge 
of the scattering and absorbing power of wakes at 
different ranges behind a vessel, and at different 
depths below the surface, would be very useful in the 
design of equipment for such methods of attack. 

26.3 ACOUSTIC WAKE RESEARCH 

The aim of current wake studies is twofold: (1) to 
explore the overall acoustic properties of wakes with 
a view to possible tactical applications, and (2) to 
advance fundamental research on the structure and 
physical constitution of wakes. The second problem 
may seem rather academic to those who are prima¬ 
rily interested in the first one. But many questions 


about wakes presented by naval tactics cannot be 
answered satisfactorily, at present, for lack of a 
thorough understanding of the physical constitution 
of wakes. Thus in the long run, fundamental re¬ 
search is indispensable for developing a comprehen¬ 
sive doctrine of the use of wakes in naval warfare. 

The solution of that fundamental problem in itself 
largely depends on acoustic measurements. Since 
wake research is still in an early stage, and since only 
incomplete observations are at hand, it would be im¬ 
practical to insist upon strict separation of these two 
aims. Experimental data frequently are relevant from 
the point of view either of tactical applications or of 
fundamental research. Accordingly, a certain shift 
back and forth between practical and theoretical em¬ 
phasis is unavoidable. 

In order to plan, execute, and interpret acoustic 
measurements on wakes, some working hypothesis 
concerning the nature of acoustic wakes must be 
used as a starting point. Three physical explanations 
of the causes of scattering and absorption of sound in 
the sea have been suggested. The scattering and 
absorbing centers have tentatively been identified 










ACOUSTIC WAKE RESEARCH 


447 



Figure 7. Close-up of USS Ringgold (DD 500), from 300 feet 












448 


INTRODUCTION 



with (1) air bubbles of widely varying size; (2) tur¬ 
bulent motion in the sea, on a scale small compared 
with the dimensions of ships; and (3) thermal in¬ 
homogeneities or irregularities in the sea, also on a 
small scale. 

Although the bubble theory of acoustic wakes now 
enjoys general acceptance, it is difficult to put it to a 
conclusive test; it has been adopted rather by de¬ 
fault of the other two explanations. It would seem 
logical, therefore, to begin by presenting the evi¬ 
dence which shows that the turbulent and thermal 
microstructure of the sea does not provide an ade¬ 
quate explanation of the acoustic properties of wakes. 
However, in order to simplify the exposition, it is pref¬ 
erable to discuss first the physical mechanism of the 
formation and dissolution of bubbles in Chapter 27 
and their acoustic properties in Chapter 28, and to 
defer the necessarily rather cursory treatment of the 
temperature and velocity structure of the sea until 
Chapter 29. The theoretical Chapters 27 to 29 com¬ 
prise the delineation of the working hypothesis which 
guides current wake research. Then the bulk of this 
volume (Chapters 30 to 33) describes the technique 
and the results of acoustic measurements made on 
wakes. In Chapter 34, the experimental data are 
interpreted in terms of the bubble theory; in other 
words, a test of the working hypothesis is under¬ 
taken. In the final Chapter 35, some conclusions 
which should be relevant in practice are drawn from 
the previous observations. Incomplete as the experi¬ 
mental foundations of some of these conclusions are, 
it appears useful to formulate some tentative general¬ 
izations as to the geometry and acoustic properties of 
wakes. Pending future research that may fill the con¬ 
spicuous gaps in our knowledge of wakes, such gen¬ 
eralizations should answer at least some of the ques¬ 
tions about wakes raised by the demands of naval 
tactics. 


Figure 8. Wake of USS Saratoga (CV3) from 4,000 
feet. 







Chapter 27 

FORMATION AND DISSOLUTION OF AIR BUBBLES 


A ir may be entrapped mechanically at the ocean 
surface and dispersed in the form of bubbles; a 
familiar example is the appearance of white caps on a 
rough sea. A great dea 1 of air is also trapped along the 
waterline of any vessel under way. Proof that such 
entrained air is capable of producing acoustic wakes 
comes from experiments on the wakes of sailing ves¬ 
sels. Probably the most copious source of bubbles in 
wakes, however, is propeller cavitation at high 
speeds. 

27.1 FORMATION OF BUBBLES BY 
CAVITATION 

When a cavity is created in water containing dis¬ 
solved air, gas enters the cavity by diffusion, and 
when the cavity collapses, this gas remains behind as 
a bubble. The process of underwater formation of 
bubbles, therefore, involves two quite different 
phenomena: (1) the mechanics of cavitation, and 
(2) the thermodynamics of diffusion and solution of 
gases in liquids. 

27.1.1 Mechanics of Propeller 
Cavitation 

The phenomenon of propeller cavitation has long 
been known to engineers. According to hydrody- 
namical theory, cavities in liquids originate when 
certain patterns of flow produce regions of negative 
pressure near propellers. Such regions are set up in 
the vortices formed near the propeller tips, provided 
that the tip speed exceeds a certain critical limit, and 
also on the back side of the propeller blade. Hence, 
it is customary to speak of tip vortex cavitation and 
blade cavitation. These theoretical deductions have 
been verified experimentally by taking high-speed 
photographs of propellers running under water, 
shown in Figures 1, 2, and 3. 

By driving a propeller-in an experimental chamber 
and observing it through a window, the process of 


cavitation can be followed visually under strobo¬ 
scopic illumination. When the speed of the propeller 
is gradually increased, bubbles are seen first to form 
at the propeller tips, from which they spiral back¬ 
ward in a long stream. Then bubbles begin to cover 
the part of the blade closest to the tips, forming a 
sheet on the blade. This phenomenon is sometimes 
described as laminar cavitation, in order to distin¬ 
guish it from the formation of larger bubbles on the 
blade face nearer to the hub, called burbling cavita¬ 
tion, which starts at still higher speeds. Physically, 
there is no sharp distinction between laminar and 
burbling cavitation, and it would be more appro¬ 
priate to classify them together as blade cavitation. 

While persistent cavities are particularly likely to 
be formed in the tip vortices and on the propeller 
blades, cavitation also may be produced around 
sharp projections on the ship’s hull, especially during 
periods of sharp acceleration of the ship. For in¬ 
stance, white foamy spots have been observed visu¬ 
ally from a launch on the superstructure of a sub¬ 
merged submarine that passed at shallow depth. The 
appearance of the white spots did not suggest the re¬ 
lease of a stream of entrapped air; hence, the spots 
were tentatively attributed to cavitation occurring 
on the superstructure. 1 This result cannot be re¬ 
garded as general, since the submarine had not been 
submerged for a long enough time to justify assum¬ 
ing that all surface air entrained during the dive 
had been dislodged by the time of the observation. 

27.1.2 Growing and Shrinking of 
Bubbles 

After a cavity has been formed in sea water which 
is saturated with air at an external pressure of 1 
atmosphere, gas begins to diffuse into the vacuum 
from the surrounding liquid. Since the diffusion con¬ 
stants for oxygen and nitrogen are nearly equal, the 
gas collecting in the cavity must have the same com¬ 
position as that dissolved in the sea water. This corn- 


449 


450 


FORMATION AND DISSOLUTION OF AIR BUBBLES 



Figure 1. Cavitating model propeller. The picture was made with a 1/30,000-sec flash. Note the heavy tip vortices, 
considerable laminar cavitation near the blade tips, and the start of burbling cavitation of the blade face near the hub. 
This is a right-hand propeller and the water is flowing from left to right. 


position differs markedly from that of atmospheric 
air because the solubility of nitrogen is twice that of 
oxygen. Accordingly, the cavitation gas consists of 
}i oxygen and % nitrogen. The quantity of gas 
which collects each second in a cavity in moving 
water is proportional to the surface area of the 
cavity and to the partial pressure of air dissolved in 
the surrounding water but is essentially independent 
of temperature and hydrostatic pressure. The con¬ 
stant of proportionality is roughly 4 X 10~ 9 mole 
per sq cm per second per atmosphere. 2 

When the cavity collapses, the gas which has dif¬ 
fused into it will be compressed, and a bubble will be 
formed with a radius such that the gas pressure in¬ 
side equals the hydrostatic pressure outside. The 
cavities formed by blade cavitation collapse so 
quickly that any air bubbles formed must be very 
small indeed. However, the cavities originating in the 


tip vortices last much longer, since the centrifugal 
force in the whirling vortex remains high for some 
time. Thus, presumably it is the tip vortex cavita¬ 
tion that is primarily responsible for most of the air 
appearing as bubbles in propeller wakes. It has been 
observed that sea water at all depths contains dis¬ 
solved oxygen and nitrogen in amounts roughly cor¬ 
responding to saturation at the surface. For this 
reason it is undersaturated with respect to a bubble 
of air or cavitation gas anywhere below the surface, 
and a bubble of either gas will gradually disappear 
as the gas reenters the water. The rate of solution 
agrees with the same simple theory of diffusion as the 
rate of accumulation of gas in a cavity; indeed, the 
facts regarding the latter process are largely inferred 
from a study of the former. The number of moles of 
air which escape each second from a bubble is ap¬ 
proximately proportional to the surface area of the 






FORMATION OF BUBBLES BY CAVITATION 


451 



Figure 2. Cavitating model propeller. Picture made with a 1/30,000-sec flash. Shows heavy tip vortices extending 
down over leading edge and fairly wide area of blade covered by burbling cavitation. This is a right-hand propeller and 
the water is flowing from left to right. 



Figure 3. Cavitating model propeller. Short sections of the tip vortices are quite clear and the development, growth, 
progress, and disappearance of individual bubbles in the cavitation on the back of the upper blade can easily be followed. 
This is a right-hand propeller and the water is flowing from left to right. 













452 


FORMATION AND DISSOLUTION OF AIR BUBBLES 


bubble and to the difference between the pressure in 
the bubble and the partial pressure of air dis¬ 
solved in the water. The constant of proportionality 
is again 4 X 10 _s mole per sq cm per second per at¬ 
mosphere. An alternative formulation, assuming a 
spherical bubble, is in terms of the rate of decrease of 
the bubble diameter per second. In water saturated 
with air at the surface, this rate increases from 
8 X 10~ 5 cm per sec at a depth of 5 meters to 
18 X 10 -5 cm per sec at a depth of 100 to 200 meters. 
Beyond these depths there is no further significant 
increase. 



RADIUS OF BUBBLE IN CENTIMETERS 


Figure 4. Rate of rise of air bubbles in still water. 

A. Rectilinear motion, spherical shape. B. Helical 
and twisting motion, flattened shape. C. Irregular. 

D. Rectilinear motion, distorted mushroom shape. 

These theoretical ideas concerning the formation 
and dissolution of bubbles have been tested in a 
series of simple experiments; 2 their agreement with 
the theory appears to be satisfactory. However, it 
remains uncertain to what extent these conclusions 
reached are applicable to the conditions prevailing 
in wakes. According to the experiments, a bubble 
0.1 cm in radius, which is the resonant size for 3 kc 
sound, should dissolve completely in about 20 min¬ 
utes. If the wake originally contains bubbles of all 
sizes up to 10~ 2 cm radius, then as the smaller bubbles 
contract, the larger bubbles also decrease in size; and 
some bubbles of the smallest size should be found 20 
minutes after the formation of the wake. In rough 
agreement with theoretical expectations, acoustic 
effects of wakes at supersonic frequencies are ob¬ 
served to persist over periods from 15 to 45 min¬ 
utes. In a wake, bubbles travel in a field of turbulent 
motion, rising gradually to the ocean surface where 


they may disintegrate; this process constitutes an¬ 
other important factor limiting the lifetime of wakes. 
The next point to be considered, therefore, is the 
buoyancy and the rate of ascent of air bubbles in sea 
water. 

27.2 BUOYANCY AND RATE OF ASCENT 

The unimpeded rise of bubbles through still water 
has been analyzed in great detail.® From this analy¬ 
sis of all available experimental data and from certain 
theoretical considerations, a curve was constructed 
which gives the rate of rise of air bubbles in water as 
a function of the radius of the bubble and is repro¬ 
duced in Figure 4. It will be noted that the velocity 
reaches a maximum at a radius of about 0.1 cm and 
varies onty slightly with the radius thereafter. Sev¬ 
eral distinct types of motion and shapes of bubbles 
have been found to be characteristic in various 
ranges of bubble radii and are shown in Figure 4. No 
exact delineation of these radius intervals can, how¬ 
ever, be made. All observers agree that for very small 
bubbles the motion is linear. For large bubbles the 
motion is also approximately linear, although some 
irregularities have been reported. A noteworthy 
feature of the velocity curve for radii up to 0.04 cm 
is that it coincides with the empirical curve for the 
rate of fall through water of spheres of specific 
gravity 2. In connection with the laboratory experi¬ 
ments on bubble screens, 4 which will be described in 
the next chapter, this relation between bubble radius 
and rate of rise has been tested empirically, and ex¬ 
cellent agreement was found over a range of bubble 
radii from 0.01 to 0.1 cm. These rates of rise of bub¬ 
bles in still water, as predicted from purely gravita¬ 
tional theory, would lead to the conclusion that all 
bubbles of acoustically effective size would reach the 
ocean surface in a time much shorter than the com¬ 
monly observed lifetime of an acoustic wake. 

However, the motion of the ship’s hull and the 
action of its propellers continually set up throughout 
the wake a strongly turbulent internal motion, which 
interferes with the streaming of bubbles toward the 
surface resulting from their buoyancy. This phe¬ 
nomenon is analogous to the transportation of sus¬ 
pended material in rivers. Most suspended material 
is heavier than water and, therefore, would settle 
out in nonturbulent flow. But through turbulence 
this material is maintained in a state of suspension. 
Similarly, in a wake the bubbles rise toward the sur¬ 
face, while turbulence counteracts this tendency. 
























BUOYANCY AND KATE OF ASCENT 


453 



Figure 5. Bow wave, hull wake, and stern wake of USS Idaho (BB42). 















454 


FORMATION AND DISSOLUTION OF AIR BUBBLES 



Figure 6. Underwater photograph of cavitation spot near bow of a PT boat traveling at 9.5 knots. 


The analogy with transport in a river is not complete, 
since the turbulence at any fixed position in a wake 
dies out gradually and the bubbles, once they have 
reached the surface, are likely to disintegrate. 

A semi-theoretical analysis of the lifetime of wakes 
has been presented which aims at finding precisely 
how much turbulence is needed in order to account 
for the observed ages of acoustic wakes. 5 In this 
work, the intensity of turbulence is measured by a 
certain empirical parameter, and it is shown that the 
theoretical lifetime of the wake passes through a 
broad but well-defined maximum if the turbulence 
parameter is increased steadily. 

This theoretical maximum has a simple qualitative 
physical explanation. While weak or moderately 
strong turbulence tends to lengthen the lifetime of a 
wake, as pointed out before, a very large degree of 
turbulence will speed the decay of a wake by in¬ 
creasing the probability of the bubbles reaching the 
ocean surface and breaking up, namely, when the 


average value of the upward components of the tur¬ 
bulent motion exceeds the speed of the rise of bub¬ 
bles with gravitational force alone. The existence of 
these opposing effects for very small and very large 
turbulence accounts for the maximum lifetime 
reached at some intermediate value of the turbulence 
parameter. The predicted maximum happens to agree 
with the average observed lifetime of acoustic wakes, 
which is from 15 to 45 minutes. Gratifying as this 
result is, there are not available any measurements of 
the intensity of turbulence in wakes, and hence the 
actual value of the turbulence parameter is unknown. 

Moveover, should the observations necessary to 
specify the value of the turbulence parameter be 
made, the analysis 3 would require some modifica¬ 
tion before an exact comparison with the observed 
lifetime of wakes could be made. In particular, the 
concentration of bubbles at the ocean surface was 
assumed to vanish, according to the premise that the 
bubbles reaching the surface are immediately de- 





ENTRAINED AIR 


455 



Figure 7. Underwater photograph of white water under hull of a PT boat traveling at 9.5 knots. 


stroyed and thus removed from the ocean. Even 
granting the validity of this physical assumption, the 
removal of bubbles cannot be expressed mathemati¬ 
cally by a vanishing bubble density. Inasmuch as the 
number of bubbles reaching the surface per unit time 
and per unit area equals the product of the bubble 
density and their average velocity upward, a vanish¬ 
ing bubble density implies a vanishing number of 
bubbles reaching the surface and thus does not cor¬ 
respond to the physical situation envisaged. In addi¬ 
tion, the decay of turbulence as the wake ages may 
also have to be considered. 

27.3 ENTRAINED AIR 

The fact that sailing ships have a conspicuous wake 
suggests that a good deal of air is trapped along the 
waterline of any vessel under way. Such air might 
materially contribute to the mass of bubbles ap¬ 


pearing in the wake of vessels propelled by engines. 
For instance, if Figure 5 could be relied on, the hull 
wake on the starboard side of the USS Idaho (BB42) 
would be even stronger than the stern wake. Of 
course, nothing is known about the extension in depth 
of the respective foam masses. 

Qualitative tests 6 showed that echoes from the 
wake of a barge towed by a tug alongside could be 
detected with an NK-1 type shallow depth recorder 
ranging downward from a launch carried across the 
wake. However, it was found that this wake was more 
acoustically transparent than the wakes of vessels 
propelled by screws and therefore probably had a 
shorter lifetime. 

These conclusions were confirmed by experiments 
in which sailing furnished the motive power. The 
ship used was a 104-ft yacht; measurements were 
made as described for other ships in Section 31.3. 
The wake when using sail was never found to be 




FORMATION AND DISSOLUTION OF AIR BUBBLES 


45G 



Figure 8. Underwater view from port quarter of a PT boat traveling at 6 knots showing propeller cavitation. 


acoustically opaque enough to blank out the bottom 
of San Diego Bay, where all the experiments were 
made. The wake thickness did not differ significantly 
from that observed in runs made with engines only, 
with the same vessel under comparable weather 
conditions; the average thickness was 12.4 ft with 
engines, and 13.0 ft under sail. As far as this scanty 
evidence goes, the geometric form of the wake seems 
to be determined primarily by the shape of the hull 
of the vessel and its speed, and it seems to be imma¬ 
terial whether the bubbles are produced by entering 
surface air or by propeller cavitation. 


A novel direct approach to the visual study of the 
subsurface structure of wakes has been made pos¬ 
sible by the recent development of underwater mo¬ 
tion pictures at the David Taylor Model Basin. This 
technique should also prove most useful for revealing 
the distribution of entrained air around the hull. 
For instance, when a small power boat passed with 
a speed of 2 to 3 knots over the underwater camera, 
mounted on the bottom in shallow water, the film 
shows a strongly foaming, shallow stern wake ex¬ 
tending backward from the hull wake over a-con¬ 
siderable distance. This wake did not reach down to 





ENTRAINED AIR 


457 



Figure 9. Underwater view from starboard quarter of a PT boat traveling at 19 knots showing propeller cavitation. 


the depth of the screw of the launch. In fact, no 
stream of bubbles could be detected as emanating 
from the screw, which was clearly visible since the 
launch approached the camera as closely as 12 ft; 
presumably a speed of only 2 or 3 knots was insuf¬ 
ficient to reach the cavitation limit. 

Figures 6 to 11 are selected frames from an under¬ 
water motion picture showing a PT boat, outfitted 
with three screws, and its wake. These pictures were 
taken in water about 40 ft deep, near the Dry Tor- 
tugas; the choice of this location was dictated by the 
need for considerable optical transparency in the 
ocean. The motion picture camera was mounted, 
slanting upward, on a steel tower firmly anchored 
on the ocean bottom, and was operated by a diver. 
The distance from the camera to the ocean surface 
was about 15 ft. 


At the left in Figure 6, a small amount of entrained 
air is visible along the water line. Moreover, a sharply 
outlined cavitation spot is conspicuous; unfortu¬ 
nately no attempt was made to ascertain what sort 
of unevenness on the hull caused this cavitation. In 
Figure 7 a large amount of entrained air is seen cov¬ 
ering the hull. Both Figures 6 and 7 were made as 
the vessel traveled at a speed of 9.5 knots; there is 
no explanation of why the amount of entrained air 
differs so greatly in these two illustrations. 

Figures 8 to 11 illustrate the progressive develop¬ 
ment of propeller cavitation as the speed increases. 
They furnish an instructive corollary to Figures 1, 
2, and 3, and show that tip-vortex cavitation caused 
by the screws of a vessel under way at high speeds 
has the same appearance as that behind a laboratory 
propeller driven by a stream of moving water. 









458 


FORMATION AND DISSOLUTION OF AIR BUBBLES 



Figure 10. Underwater view from starboard quarter of a PT boat traveling at 27 knots showing propeller cavitation. 




ENTRAINED AIR 


459 



Figure 11. Underwater view from astern of a PT boat traveling at 36 knots showing propeller cavitation 





Chapter 28 


ACOUSTIC THEORY OF BUBBLES 


T he rigorous treatment of the acoustic char¬ 
acteristics of bubbles, especially of the cumula¬ 
tive effects of a multitude of bubbles, requires a 
great deal of rather advanced mathematics. For a 
comprehensive exposition of these theories, reference 
must be made to several monographs on the sub¬ 
ject. 1-4 In this chapter only the principal features 
of the problem will be sketched, primarily with a 
view to the later elementary interpretation of the 
acoustic properties of wakes in Chapter 34. Actual 
wakes have such a complicated structure that many 
physical and mathematical refinements incorporated 
in the rigorous treatment of certain ideal cases have, 
at present, only academic interest. 

The first two sections of this chapter deal with the 
acoustic properties of individual bubbles. In the 
third section, the combined acoustic effects of many 
bubbles are discussed, and the results are applied to 
the evaluation, from laboratory experiments, of cer¬ 
tain physical constants — acoustic cross sections, 
damping constants, which cannot as yet be pre¬ 
dicted from pure theory. 

28.1 SCATTERING BY A SINGLE IDEAL 
AIR BUBBLE 

For application to wakes, only those air bubbles 
need be treated whose radius R is very small com¬ 
pared with the wavelength A of sound in water, or 

X 2irR 

R«- or v = ~ -«1, (1) 

zw X 

where g is the ratio of the bubble circumference to the 
wavelength. Then the pressure amplitude of the in¬ 
cident sound wave can be regarded as constant in¬ 
side the bubble and in its immediate vicinity. De¬ 
noting this amplitude by A, the pressure P» of the in¬ 
cident wave can be described by 

P 0 = Ae 2 " ifl . (2) 

Although the effect of the sound wave impinging 
upon the air bubble is a rather complex one, the re¬ 


sulting phenomena may be classified under two prin¬ 
cipal headings. 

First, the periodically variable pressure of this in¬ 
cident sound wave produces a forced vibration of the 
air inside the bubble, which reacts on the surround¬ 
ing water and produces in turn an emission of sound 
waves from the bubble. This secondary sound wave 
is spherically symmetrical for all practical purposes. 
Such a process of transforming an incident wave into 
waves of different pressure distributions is commonly 
known as scattering. The concept of scattering refers 
solely to the process of redistribution of sound energy 
— in other words, it is understood that none of the 
sound energy is converted into other forms of energy. 
Actually, scattering by a bubble is always accom¬ 
panied by conversion of part of the impinging sound 
into heat by any one of a number of processes. These 
effects are described together under the name of ab¬ 
sorption of sound. In the present section, scattering 
by a single bubble will be treated as though it were 
possible to produce this phenomenon apart from ab¬ 
sorption; therefore, the term “ideal air bubble” has 
been used in the title of Section 28.1. 

If the incident sound wave is a plane wave, the 
rms sound intensity / 0 , or the average rate at which 
sound energy crosses a unit area placed perpendicular 
to the sound beam, is according to equation (56) in 
Chapter 2 


where A is the complex pressure amplitude, c is the 
velocity of sound in sea water, and p is the density of 
sea water. These waves excite vibrations of the air in 
the bubble and indirectly excite pressure waves in 
the surrounding water. In order to compute rigor¬ 
ously the possible types of vibration, the method of 
normal modes of vibration would have to be applied 
(see Section 27.1). It can be shown that of the various 
modes of vibration only the spherically symmetrical 
ones are significant in the present analysis, and that 
the other modes, corresponding to directional pat- 


460 


SCATTERING BY A SINGLE IDEAL AIR BUBBLE 


461 


terns of scattering, can be neglected. 1 ’ 2 This princi¬ 
pal mode of spherical symmetry causes the scattered 
sound to be a spherical wave centered upon the 
bubble, which can be described by the formula, 


r 


(4) 


where p s is the pressure of the scattered wave, r is the 
distance from the center of the bubble to the scat¬ 
tered wave, and B is the complex pressure amplitude 
of the scattered wave at unit distance; then B/r is the 
pressure amplitude at the distance r. The intensity 
of the scattered wave /, at a distance r is then, 


/. 


[£ | 2 

2 cpr- 


(5) 


The problem now is to calculate the intensity I s of 
the divergent scattered wave from the intensity 7 0 of 
the plane incident wave. At this point it is convenient 
to introduce the cross section a s of the bubble for 
scattering of sound, which is defined by 


0’s = 


4tt | B 

\A\‘ 


( 6 ) 


The physical meaning of a s is simple. As the intensity 
of the scattered sound is given by equation (5), the 
total scattered energy at this distance r from the 
bubble center is 4irr' 2 I s . Thus by combining equations 
(3), (5), and (6), 

Air r-I s = aJo. (7) 


Hence, the sound energy flowing through an area a 3 
perpendicular to the incident sound beam is equal to 
the total energy scattered by the bubble in all direc¬ 
tions. The bubble itself exposes to the incident wave 
the cross-sectional area irR-. If all energy intercepted 
by this area would be converted into scattered sound, 
the rate of scattering by the bubble, or the energy 
scattered per unit time, would be irR-I 0 . 

Evidently, whether the scattered energy is smaller 
or greater than the energy geometrically intercepted 
bv the bubble will depend on the ratio a a /irR' 2 . The 
incident wave excites pulsations of the bubble, which 
are forced vibrations of the frequency / of the incident 
sound. They will interfere with free vibrations of the 
bubble at resonance, the frequency f T of which will be 
computed presently. As is well known, the forced 
vibrations of any mechanical system become very 
intense if /is near the frequency f T of the free vibra¬ 
tions characteristic of this particular system. In this 
case of resonance the scattered energy can become 
considerably greater than wR-R) thus the scattering 


cross section cr s may far exceed the geometric cross 
section -n-R 2 of the bubble. 

In order to find cr 3 , or Air [Bj 2 /|A.| 2 , the radial 
velocity of the bubble surface vr will be computed in 
two different ways, following the treatment in a re¬ 
port by CUDWR. 5 On one hand, there is a hydro- 
dynamical boundary condition which the air in the 
bubble has to satisfy during its vibrations, namely 
that the instantaneous gas pressure inside the bubble 
must be equal to the external acoustic pressure. On 
the other hand, there is a relation between pressure 
and volume (or pressure and radius) which the air in 
the bubble has to satisfy during its vibration, ac¬ 
cording to the principles of thermodynamics. 

If the vibrations are so rapid that there is no heat 
exchange between the bubble and its surroundings, 
it may be assumed that the pulsations are adiabatic 
changes of state. For such changes, it is found from 
the first law of thermodynamics and from the equa¬ 
tion of state of an ideal gas that PV 1 ' remains con¬ 
stant during the pulsations, where P is the air pres¬ 
sure inside the bubble, V the volume, and y the ratio 
of the specific heats, which for air is 1.4. Denoting 
by P o the average hydrostatic pressure in the water, 
by To and R 0 the volume and radius of the bubble in 
the state of equilibrium, the condition for adiabatic 
pulsations with small departures dV and dP from the 
equilibrium volume and pressure can be obtained by 
differentiating the relation PV y = constant: 

dP _ _ dV J_ dP _ _ y_ dV 
Po ~ _ 7 Vo Po dt~ ~ Vo dt ' 

By expressing the volume of the bubble in terms of 
its radius P, it is found that 

4 , dV , dR dR 

V ‘ = V R °' ~it ~ irR °~di ’ 

If pi denotes the acoustic pressure and A, the 
pressure amplitude inside the bubble, the forced 
vibrations of the air in the bubble are described by 

V i = Aie 2rifl , if A*?’*. (10) 

dt 


This internal acoustic pressure p, is to be identified 
with the excess gas pressure inside the bubble, dP, 
which appears in equation (8). Hence, by substitu¬ 
tion of equations (9) and (10) into equation (8) it 
follows that 


dR 


Vr = 


2irifR,A; 


‘2 wift 


( 11 ) 


dt SyPo 

The next step is to compute the amplitudes A 





402 


ACOUSTIC THEORY OF BUBBLES 


and B from the given pressure amplitude of A of the 
incident sound wave and from the hydrodynamical 
boundary conditions. These boundary conditions are 
formulated in Section 2.6.1. They require that the 
pressure p and the component v R of the particle 
velocity normal to the surface have no discontinuity 
at the surface; the continuity of v R is equivalent to 
the continuity of (1 /p)dp/dR, which was required in 
Section 2.6.1. 

While the pressure inside the bubble is given 
by equation (10), the outside pressure is the sum of 
the incident wave, expression (2), and the scattered 
wave, expression (4). The pressure of the scattered wave 
p s at the surface of the bubble is found from equation 
(4), with r set equal to R 0 ■ Because of assumption (1), 
namely that the linear size of the bubble is small com¬ 
pared with the wavelength X, the term r/X in the 
exponent of equation (4) is much smaller than unity 
in the vicinity of the bubble and, therefore, p s and its 
derivative can be replaced approximately by 


P> 


^Ro 





dp* 

dr 


~ 2 e 2 " 7 '. (12) 


Then the continuity of the pressure at the surface 
of the bubble is expressed by 

Pi = Po + Ps- 


If the expressions (2), (10), and (12) are substituted 
into this equation, and the common factor e 2nlt can¬ 
celed, the following equation is found to apply at the 
bubble surface during the small oscillations usually 
encountered in practice: 

B 2iri 

= (13) 

i to A 

The normal component of the velocity must also 
be continuous at the bubble surface. From equation 
(11) the normal component of the fluid velocity in¬ 
side the bubble is known. Its value outside the bubble 
can be derived from equation (12). The relation be¬ 
tween the fluid velocity and the pressure gradient 
dp/dr in a certain direction is, according to the argu¬ 
ment in Section 2.61, 


dv R dp 
dt ~ ~ dr ‘ 


(14) 


By substituting for dp/dr the derivative of the pres¬ 
sure of the scattered wave from equation (12) with r 
set equal to R 0 and integrating over dt, equation (14) 
is transformed into 


Bi 

2irfpRl 


e 


2 wifi 


(15) 


To this equation the amplitude A of the incident 
wave does not make any contribution, because the 
wavelength is much larger than the size of the bubble; 
since the pressure gradient dp 0 /dx is uniform in the 
vicinity of the bubble, the velocity corresponding to 
this uniform pressure gradient is a motion of the 
entire bubble to and fro rather than an expansion and 
contraction of the bubble. The continuity of v R can 
now be formulated, according to equations (11) and 
(15), by 


B _ 2tt/7iV4 , 
2wfpRo SyP 0 


From equations (13) and (16) A. can be eliminated 
and a relation between A and B obtained; if the 
subscript 0 is omitted from R 0 , the bubble radius in 
equilibrium, then 


3tPq 2-iriR 

Air-f-pR- ~ + X 


(17) 


By introducing now the abbreviation f r , defined by 



the physical meaning of which will soon become ap¬ 
parent, equations (17), and (18) and (1) may be 
combined to give the result 



(19) 


In order to obtain the scattering cross section a s of 
the bubble from equation (6), \A | 2 and | B\- have to be 
computed from 


A\* = AA*,\B\* = BB*, (20) 


where A* and B * are the complex conjugates of A 
and B respectively. According to equation (19), 



( 21 ) 


Finally, from equations (6), (19), (20), and (21), 
47 rR 2 

- • ( 22 ) 

+ T 

For a bubble of a definite radius R, the scattering 
cross section a a has its peak value if/ equals f T ; it is 
then said that the incoming wave is in resonance 
with the pulsations of the bubble, and hence f r is 



v R = 











SCATTERING BY A SINGLE IDEAL AIR BUBBLE 


463 



Figure 1. Scattering cross section for an ideal bubble. 

called the resonance frequency for the bubble of 
radius R. 

A plot of cr s / ttR' 2 as the function of y = 2ttR/\ 
= 2-irRf/c is shown in Figure 1, the outstanding 
feature of which is a sharp peak. This maximum cor¬ 
responds to the resonance value y r or according to 
equation (18) 

T]r = = - \/hEl = 1.36 X 10- 2 , (23) 

c c r p 

if P 0 is atmospheric pressure and c is the sound 
velocity in sea water at 60 F. Thus, at resonance, <r s 
is enormously greater than the geometric cross sec¬ 
tion of the bubble; specifically 

— = f-Y = 2.16 X 10 4 , (24) 

7 tR 2 \r)r' 

where ov is the value of <r s at resonance. Equation 
(24) can also be expressed in the form 



TT 


While equations (24) and (25) must be considerably 
modified for an actual bubble, as shown in the next 
section, the phenomenon of resonance is neverthe¬ 
less responsible for the great efficiency of bubbles as 
scattering agents. Moreover, the resonant frequency 
found from equation (18) is correct for a wide spread 


of bubble sizes. This equation has been confirmed by 
observations at low frequencies, between 1,000 and 
6,000 c per sec, 6,7 and also at high supersonic fre¬ 
quencies, between 20 and 35 kc. 7 In each case a 
single bubble was placed in the sound field, and the 
sound frequency determined at which the bubble 
oscillated most violently. The radius of the bubble 
was then measured either with a microscope, or for 
the larger bubbles by measurement of the volume 
of air in the bubble. The values of the resonant 
frequency f r found in these measurements for bubbles 
of air, hydrogen, and oxygen in water at different 
temperatures agreed with equation (18) within the 
experimental error of about 5 per cent. Thus within 
the range from 1 to 50 kc equation (18) may safely 
be used to predict the resonant radius of bubbles in 
water. Values computed from this equation are 
given in Table 1. 


Table 1. Resonant radius for air bubbles in water. 


Frequency in kc 
Wavelength in centimeters 

1 

150 

5 

30 

20 

7.5 

50 

3 

Pressure 






Depth of 





Atmospheres 

water 






in feet 





1 

Surface 

0.33 

0.065 

0.016 

0.006 

2 

35 

0.47 

0.093 

0.023 

0.009 

5 

140 

0.73 

0.15 

0.037 

0.015 

10 

300 

1.04 

0.21 

0.052 

0.021 


For very small bubbles, with radii less than 10 -3 
cm, surface tension becomes important and the com¬ 
pressions and expansions of the gas in the bubble be¬ 
come isothermal instead of adiabatic. No observa¬ 
tions for such small bubbles are available, but a 
theoretical analysis 5 shows that equation (22) is 
still valid provided that / r is defined by the equation 



(26) 

4T 


9 ~ 1 + SRP 0 ’ 

(27) 


the quantity T is the surface tension of the gas- 
liquid surface, and other quantities have the same 
meaning as in equation (18). Equations (26) and 
(27) should not be used for bubbles of radii greater 
than 10~ 3 cm. 

Equation (22), in addition to predicting the im¬ 
portance of resonance, also gives correctly the scat- 



















































ACOUSTIC THEORY OF BUBBLES 


464 


tering coefficient for frequencies considerably greater 
than the resonant frequency / r . Since r? is less than 
one, rj 1 in equation (22) may be neglected when the 
ratio fr/f is much greater than one. Consequently, the 
scattering cross section for low-frequency sound may 
be written approximately as 

<7 S = 4t ri? 2 ( j ) 4 = 4t tR* 4 • (28) 

This equation is known as Rayleigh’s law of scatter¬ 
ing for long-wave radiation. It will be remembered 
that in optics Rayleigh’s law explained the blue color 
of the sky, as the resonant frequencies characteristic 
of the atmospheric gases oxygen and nitrogen are far 
greater than the frequencies of visible light. Hence, 
the shorter (blue) waves of sunlight are scattered 
more strongly than the longer (red) waves and reach 
our eyes with greater intensity. Equation (28) is also 
applicable to the high-frequency sound commonly 
used in echo ranging provided that the bubble 
radius R is very small; if R is less than 10 -3 cm, how¬ 
ever, f r is given by equation (26) instead of by equa¬ 
tion (18). 

28.2 SCATTERING AND ABSORPTION 
BY AN ACTUAL BUBBLE 

So far, the attenuation of sound resulting from the 
absorption of sound energy during the pulsation of 
the bubble has been neglected. The existence of such 
an effect is a direct consequence of the second law of 
thermodynamics, which implies that energy must be 
extracted from the sound field and dissipated into the 
surrounding water in the form of heat, in order to 
maintain the forced pulsation of the bubble against 
the internal friction of the bubble-water system. In 
other words, it is thermodynamically inadmissible 
to treat the pulsation of the bubble as if it were a 
strictly adiabatic process; therefore it becomes neces¬ 
sary to amend the analysis given in the preceding 
section for an ideal bubble. 

This task is accomplished by adding to equation 
(13), which expressed the continuity of pressure at 
the bubble surface, a certain term which takes into 
account the frictional force modifying the behavior 
of an actual bubble. Moreover, the exchange of heat 
between bubble and water by conduction neces¬ 
sitates a modification of equation (16),which formu¬ 
lated the continuity of velocity at the bubble surface. 
The treatment of the case of the ideal bubble im¬ 
plicitly assumed that the pulsations are thermo¬ 
dynamically reversible; that is, the work put into the 


bubble during compression was supposed to be equal 
to the work done by the bubble during expansion. 
Actually, there is heat exchange between bubble and 
water, but the pulsations are too rapid to permit a 
complete leveling of temperature at every instant of 
the cycle. Thus there prevails a continual change of 
state which is somewhere between the adiabatic and 
isothermal case. 

It is not difficult to see that under such circum¬ 
stances the pulsation of pressure cannot be in phase 
with the pulsation of volume. While the bubble is be¬ 
ing compressed, the temperature rises steadily; as 
soon as the rise of temperature becomes appreciable, 
heat conduction begins to operate and the bubble 
tends to cool off even before expansion has started. 
When the minimum volume is reached, the tempera¬ 
ture will be decreasing as heat flows from the bubble 
into the water. Consequently, the temperature maxi¬ 
mum will be reached some time before the bubble has 
been compressed to its minimum volume. Likewise, 
since the gas pressure is proportional to the tempera¬ 
ture, the maximum pressure will not be attained 
simultaneously with the minimum volume, but some 
time before. Thus there exists a phase shift between 
pressure and temperature on one hand, and volume 
and radial velocity of the bubble on the other hand. 
For resonant bubbles at frequencies of 100 kc or less, 
this effect is taken into account 4 by inserting a com¬ 
plex factor 1 — pi in the right-hand side of equation 
(11), where P is a positive constant much smaller 
than one. 

The two equations of continuity, (13) and (16), 
must therefore be replaced, for an actual bubble, by 
the following ones: 

„ dR 

Po + p s — Pi = — Ci — » (29) 

at 


or 


and 


.4 + B/R 0 - 2 rikB -.4, = ~~ , 

Zttj pA(, 

B 2rfRoA s 

9/ n2 = o p C 1 “ 00 ’ 

2irfpR u 3yRo 


(30) 


In equation (29) Ci is a constant measuring the 
effect of friction, which is assumed to be proportional 
to the radial velocity dR/dt of the bubble. The term 
CidR/dt represents the net pressure on the bubble, 
which is positive when the bubble is contracting 
(■ dR/dt < 0); hence, the correction term appearing on 
the right side of equation (29) must carry a minus 
sign. 






SCATTERING AND ABSORPTION BY AN ACTUAL BUBBLE 


405 


By proceeding exactly as in Section 28.1, the fol¬ 
lowing relation is found instead of equation (19): 


B = 


RA 


71 i 

7 2 ’ i + 0 s 



- 

1 + ^ 2 Cp V 


-) 

Cpt)/ 


(31) 


If one neglects /3 2 compared to one and defines 

fr 

8 = p + t, + — , (32) 

/' CPTl 

equation (31) becomes 

RA 

B = JV 2 -x-- • (33) 

\p ~ / l6{f ’ Ro) 

Substituting this expression into equation (6), the 
cross section for scattering by an actual bubble can 
readily lie evaluated: 


4t tR 2 



It will be noted that equation (34) is identical with 
equation (22), which was derived for an ideal bubble, 
except that <5 2 has replaced rf in the denominator. 
This change affects only the magnitude of the scat¬ 
tering cross section near resonance. Thus the fre¬ 
quency of resonance anti the scattering cross section 
at frequencies far from resonance are correctly given 
by equation (22), in agreement with the statements 
made in the previous section. 

The knowledge of the scattering cross section does 
not provide all the information that is wanted in the 
case of an actual bubble, as the incident flux of 
energy is reduced both by scattering and absorption 
of sound. Calling the sum of scattered and absorbed 
energy the extinguished energy, an extinction cross 
section a e can be defined by 


<Te 



(35) 


where F, is the total energy extinguished by the 
bubble per unit interval of time and 7 0 is the intensity 
of the incident sound energy. The quantity F, is equal 
to the work done, per unit interval of time, on the 
bubble by the incoming sound beam; this extin¬ 
guished energy comprises both absorbed and scat¬ 
tered energy. Hence, F e is equal to 


F e = p 0 


dV 

Tt 


where the bar means the time average ; p 0 is the pres¬ 
sure of the incident sound wave, and V is the volume 
of the bubble. 

To evaluate equation (3G) it is simplest to use real 
quantities. According to equation (2), 

Po = Ae 2Hft . 

Since the initial phase may be chosen arbitrarily, let 
A be real, and let the sound pressure and sound 
velocity be represented by the real parts of the ex¬ 
pressions developed above. Then 

Po = A cos 2 wft. (37) 

From equations (9) and (15) it follows that 


(IV , 2 iB 2 

— - AttRIvr =-— e . 

dt fp 


Here again only the real part of the entire expression 
is to be taken. In order to find this real part, split B 
into its real part B R and its imaginary part iB 1 , and 
express e 2irlf ' in terms of its real and imaginary parts: 

dV 2 i „ 

— = —— (B R -\- iB 1 ) (cos 2-rrft -f- i sin 2-irft) 
dt -fp 

2 

= — l(B r cos 2 tt ft + B r sin 2rrft) (38) 

-fp 

+ i (B 1 sin 2-rrft — B R cos 2 rrft)']- 

If equation (37) and the corresponding real part of 
equation (38) are substituted into equation (3(i), it is 
found that 


2 A 


F e = - — (cos 2rrft) (B t cos 2rrft + B R sin 2rrft), (39). 
fp 

where the bar denotes an average over many cycles 


Since 

and 


COS 2 (2rrft) = - 


(COS 2rrft) (sin 2rrft) = 0, 
equation (39) becomes finally 

AB*_ 

Ip 

According to equation (33), 

RA8 


F„ = - 


B t = 


(M‘ 


+ 5 2 


Hence, equation (40) assumes the form 



(40) 


(41) 


(42) 


(30) 













466 


ACOUSTIC THEORY OF BUBBLES 


By combining this expression with equations (3) and 
(32), the cross section for extinction is finally ob¬ 
tained : 



The extinguished energy is obviously the sum of 
the scattered and absorbed energy. Therefore, the 
absorption cross section <r a of the actual bubble can 
be defined by the relation 

(7s T" CT (l (44) 

and is thus found to be, from equations (34) and (43), 



Note also the simple relation 



0 5 10 15 20 25 30 35 40 


FREQUENCY IN KC 

x Values found from oscillation of a single bubble 
O Values found from transmission through bubble screen 

- Adopted values of 8r 

-Theoretical curve for air bubbles 

. Theoretical curve for ideal bubbles 


0’s 


(Te 


V 

8 ' 


(46) 


Figure 2. Damping constant at resonance. 


A word must be said now about the function 8, de¬ 
fined in equation (32). If /3 and Ci are put equal to 
zero, for the case of an ideal bubble, 8 reduces to tj, 
and it is seen that equation (22) is indeed the correct 
limiting form of equation (34). Numerical values of 
/3 and Ci can be derived by an analysis of the several 
physical processes known to contribute to the absorp¬ 
tion of sound by the bubble — for instance, heat con¬ 
duction, viscosity, surface tension, and other proc¬ 
esses. There are also methods for determining 8 


<r sr 2 

4 tR 2 8* 


(47) 


where S r is the resonance value of 8 shown in Figure 
2. It (7 S (/) denotes the cross section for any non¬ 
resonant frequency, it follows from equations (40) 
and (47) that 



empirically from certain observations which will be 
discussed. Inspection of Figure 2, which shows the 
damping constant at resonance as a function of fre¬ 
quency, will reveal that the predicted values of 8 are 
much smaller than the observed ones. This discrep¬ 
ancy indicates that some relevant physical processes 
must have been overlooked in the theoretical anal¬ 
ysis of the absorption effects. Hence, theoretical 
evaluation of ft and C i, although carried out else¬ 
where, 5 will be omitted from this review, and the 
empirical values of 8 will be used for the interpreta¬ 
tion of the acoustic properties of wakes to be given 
in Chapter 34. 

The physical significance of 8 can best be visualized 
by plotting o- s /4t tR 2 against f/f r . A resonance curve, 
similar to Figure 1, is thus obtained. The peak value 
ol this graph is, according to equation (34), 


Over a narrow range of frequencies near the peak of 
the resonance curve, 8(f,R) in the denominator of 
equation (48) may be replaced by its resonance value 
8 r , and f r /f is very close to one. Hence, using the 
abbreviation q = f r /f — 1, 

(4 - i) = q\q + 2)2 = 4 9 2 , (49) 

and equation (48) becomes approximately 

*.(/) 1 


in other words, for any given small departure q from 
the resonance frequency, the decline of a s from its 
peak value is sharper for greater values of \/8 r or for 
smaller values of 8 r itself. Thegreater the sharpness of 





























SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 


467 


the resonance peak is, the smaller is the damping of 
the pulsation of the bubble. Therefore, 8 r is com¬ 
monly called the damping constant. 

28.2.1 Measurement of Damping 
Constant 

The simplest, most direct way to determine the 
damping constant 8 r is to measure the sharpness of 
the resonant peak for a bubble in a sound field. Such 
measurements have been carried out for bubbles in 
fresh water. In one case, 7 the amplitude of oscillation 
of a single bubble was observed as the sound fre¬ 
quency was slowly varied. Since a s is proportional to 
the square of the amplitude of oscillation, a plot of 
these observations yields 8 r directly. Values of S r were 
found by this method for bubbles of hydrogen and 
bubbles of oxygen, but no systematic difference was 
found between these two gases. 

In another case, the transmission loss through a 
screen of bubbles all of the same size was observed. 8 
To produce this screen, six small microdispersers ar¬ 
ranged in a line in a laboratory tank were used to 
produce a stream of bubbles 10 ft below the surface 
of the water. These bubbles were normally inter¬ 
cepted by a hood, which could, however, be swung 
to one side for about one second to allow a pulse of 
bubbles to rise to the surface. Since the larger bubbles 
arrived at the surface first, and the smaller ones at 
progressively later times, the bubbles near the surface 
at any one time were of nearly equal radii. The trans¬ 
mission loss in decibels of sound at a constant fre¬ 
quency crossing this screen was then proportional to 
ov for a single bubble; from a plot of the transmission 
loss against bubble radius, a value of <5 r could then be 
determined. A typical set of observed curves ob¬ 
tained with this technique is reproduced in Figure 4. 
In analyzing these data, account was taken of the 
variation of 8 with bubble radius so that points some 
distance from resonance could be used as well as 
those close to resonance. 

The values of 8 r found by these two methods are 
plotted in Figure 2. The dashed line curve shows the 
theoretical value of 5 r for air bubbles in water, if Ci is 
set equal to zero, and values of /3 are taken from 
reference 5. It is evident that at the higher fre¬ 
quencies the observed values are much greater than 
the theoretical values; this discrepancy has already 
been noted above. The values of 8 r found from a single 
bubble, which are shown as crosses, are somewhat 
greater than those determined from the transmission 


loss of sound through a bubble screen, plotted as 
circles in Figure 2. In the former set of measure¬ 
ments, the bubble was not free, but was caught on a 
small wax sphere fastened to a platinum thread, 
which oscillated to and fro as the bubble expanded 
and contracted. Since the damping constant may 
have been increased in this arrangement over its 
value for a free bubble, these values cannot be relied 
upon. Thus, the solid line of best fit, shown in Figure 2 
is based at high frequencies on the values found with 
the screen of freely rising bubbles. Confirmation of 
these observed values of 8 r is found in the next sec¬ 
tion, where the observed data on scattering and 
absorption of sound by bubble screens are shown to 
be in moderately good agreement with the theoretical 
values based on equations (34) and (43) and on the 
empirical curve of 8 r in Figure 2. For comparison with 
the observed values, the damping constant 8 r com¬ 
puted for an ideal bubble resonating in water at at¬ 
mospheric pressure is shown as a dashed line in the 
figure; the value plotted is taken from equation (23). 

28.3 SOUND PROPAGATION IN A 
LIQUID CONTAINING MANY BUBBLES 

The results derived in the preceding sections for a 
single bubble are only the first step toward the solu¬ 
tion of the general problem, the propagation of sound 
through a medium containing many bubbles. This 
problem is complicated because the external pressure 
affecting each bubble is the sum of the pressure in 
the incident sound wave and the pressures of the 
sound waves from all the other bubbles. While the 
mathematics of the problem is complicated, the gen¬ 
eral results to be anticipated can be presented simply. 

28.3.1 General Theory 

First, the presence of the bubbles will affect the 
nature of the medium through which the sound wave 
is progressing. If the bubbles are spaced much closer 
to each other than the wavelength, the sound velocity 
will be appx-eciably affected by the presence of the 
bubbles, which alters the compressibility of the 
medium. In addition, the sound velocity will have a 
small imaginary part, resulting from the absorption 
and scattering of sound, and giving rise to an ex¬ 
ponential drop of the sound intensity with increasing 
distance of travel through the aerated water. Thus a 
sound wave can be reflected, refracted, and atten¬ 
uated as it passes through water containing bubbles. 



ACOUSTIC THEORY OF BUBBLES 


468 


On this picture the sound wave behaves as though 
it were proceeding through a homogeneous medium, 
in which the sound velocity is a smooth complex 
function of position. 

Secondly, this picture must be supplemented to 
take scattering into account. The sound waves sent 
out from the different bubbles produce scattered 
sound, which goes out in all directions. This scattered 
radiation may be regarded as resulting from the fact 
that in a random collection of point scatterers the 
number of scatterings per unit volume is never 
constant from one region to another, but shows 
statistical fluctuations. A theory of the scattering 
of light in air is given along these lines in a well- 
known text on statistical mechanics. 9 More simply, 
the intensity of scattered radiation may be regarded 
as proportional to the average squared pressure re¬ 
sulting from all the individual bubbles. As the 
bubbles move around, the relative phases of their 
scattered wavelets will vary widely, and constructive 
and destructive interference will be equally likely. 
With this picture, the average squared pressure may 
be regarded as simply the sum of the squares of the 
pressures in each of the scattered wavelets. 

In ship wakes the number of bubbles in a small 
volume is rarely sufficiently great to produce reflec¬ 
tion and refraction of sound waves. The gradual at¬ 
tenuation of the incident sound beam and the scat¬ 
tering of sound energy in all directions by each 
bubble individually are therefore the two effects of 
greatest interest. 

The preceding discussion is, of course, not very 
rigorous. The residts stated here have been proved 
rather generally, however, in an elegant solution to 
the general problem. 3,4 This analysis makes certain 
assumptions, the most important of which are that 
the bubbles have diameters much smaller than the 
wavelength of the incident sound, and that the 
average distance between bubbles is much larger 
than their dimensions. The solution, as a result of its 
physical generality, is of considerable mathematical 
complexity, and therefore will not be reproduced 
here. But the mode of approach used in this general 
theory will be briefly sketched. 

The chief feature of this theory is its use of con¬ 
figurational averages. Different bubbles may be almost 
anywhere within a certain region. For each distribu¬ 
tion of bubbles the sound pressure p at a given time 
will have some definite value. If now an average value 
of this pressure is taken for all possible positions of 
the different bubbles, a configurational average of p, 


denoted by <p >, results. Thus is usually not equal 
to the time average of p, since this time average 
vanishes because of the oscillations of p between 
positive and negative values. Similarly, <p 2 > may 
be defined as the configurational average of p-. 

The simplified picture presented at the beginning 
of this section may be given a precise meaning in 
terms of these configurational averages. The quantity 
<p> is found to obey the wave equation in a 
homogeneous medium in which the complex sound 
velocity is a function of position. This configurational 
average acts in general as the pressure from a re¬ 
fracted sound wave. Thus <p> gives rise to a trans¬ 
mitted wave; after leaving the scattering region, this 
transmitted wave bears a definite phase relationship 
to the incident wave. 

For any particular configuration, the value of p 
may differ from <p> . A measure of this difference is 
provided by the mean square value of p — <p>, 
which is equal to <p 2 > — <p> 2 . The analysis 
shows that this difference is simply the sum of the 
squares of the pressures in the sound wave sent out 
from each of the bubbles. These additional terms 
therefore represent just the scattered sound, includ¬ 
ing sound that has been scattered several times. 
Thus the intensityat any point, which is proportional 
to p-, is on the average the sum of two terms; the 
first term <p> 2 represents the coherent wave, 
propagating through a homogeneous medium in 
which the sound velocity changes in some way with 
changing position. The second term, <p 2 > — <p> 2 
represents the sum of the scattered waves from each 
bubble. At any one time the value of p 2 , even when 
averaged over a few cycles, will usually differ from 
the sum of these two terms, but as the configuration 
of bubbles changes, the time average of p 2 should ap¬ 
proach the configurational average of p 2 . In most 
practical situations a period of several seconds is 
usually sufficient to bring the time average of p 2 
close to the configurational average. If, then, 
averages are taken over time intervals of several 
seconds, the simplified picture presented at the be¬ 
ginning of the section may be taken as correct. 

When the average distance between the bubbles 
becomes very small, or, in other words, as the 
average number of bubbles per unit volume becomes 
very large, this simplified picture becomes inadequate. 
In this case, another term must be included in <p 2 >, 
in addition to the two terms representing the re¬ 
fracted (coherent) wave and the scattered (incoher¬ 
ent) waves. This term is difficult to interpret, but 



SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 


469 


contributes to the scattered sound and appears to be 
due to interference between different scattered wave¬ 
lets. It is not easy to determine the precise point at 
which this term becomes important, but it can be 
shown to be negligible, for resonant bubbles, pro¬ 
vided the attenuation per wavelength is less than a 
few decibels. This is the same condition that must be 
satisfied if the change which resonant bubbles pro¬ 
duce in the sound velocity of the medium is to be 
relatively small. Since this condition appears to be 
satisfied in observed wakes, this additional term will 
therefore be neglected in the following derivation of 
practical formulas for the attenuation, scattering, and 
reflection of sound by water containing bubbles. 


28.3.2 Transmission 

The type of analysis developed in the preceding 
section will now be applied to find the transmission 
loss through a region containing bubbles. It will first 
be assumed that within this region all the bubbles 
are of the same size. In each cubic centimeter there 
are assumed to be n bubbles; n may vary from point 
to point within the region. If I is the intensity in the 
incident sound beam, the rate at which sound energy 
is extinguished from the beam by each bubble will 
be aj, according to equation (35). Let 1(0) be the 
intensity at the point where the beam enters the 
region containing bubbles and let I(r ) be the in¬ 
tensity after the beam has penetrated a distance r 
through the region; r is measured along a sound ray. 
The increment of I(r) after passing an infinitesimal 
distance dr is, of course, negative and has the value 

dl = — n(r)a,I(r)dr. (51) 

By integration of equation (51) over the path fol¬ 
lowed by the sound, it is found that at any distance ry 

7(n) = l(0)e~° e $'*<**' = I(0)e^' N(n) , < 52 ) 

where N(r x ) is the total number of bubbles in a 
column of length ry and unit cross section. If ri is set 
equal to w, the total thickness of the region, equa¬ 
tion (52) gives the total extinction produced by the 
bubble screen, or the attenuation as it is usually called 
in underwater sound work. Expressing the attenua¬ 
tion on a decibel scale, equation (52) is equivalent to 

10 log T ; = 10 X 0.434 X n<T,w = K e w, (53) 
I(w) 

where n is the average bubble density in the screen, 
defined by 



The quantity K, in equation (53) is usually called 
coefficient of attenuation, which is conventionally 
given in units of decibels per yard. Since n and <r,. 
are usually expressed in units of cm -3 and cm'-, re¬ 
spectively, and since there are 91.4 cm to the yard, 
/v, in decibels per yard becomes 


K e = 396.8,*<7 C . (55) 

The attenuation coefficient K e is rather easy to 
determine by acoustic measurements either of a 
wake (see Chapter 32) or of a bubble screen produced 
in the laboratory (see Section 28.2). Since a,, is known 
for resonant bubbles from the experimental deter¬ 
mination of the damping constant <5 r already described 
in Section 28.2, the bubble density n can be computed 
from K e and a e by equation (55), on the assumption 
that only bubbles of resonant size are present. How¬ 
ever, among copious masses of bubbles, as found in 
wakes, there will usually be a wide dispersion of 
bubble sizes. It is important, therefore, to evaluate 
the attenuation produced by such nonhomogeneous 
bubble populations. 

Let the number of bubbles per cubic centimeter 
with radii between R and R + dR be denoted by 
n(R)dR, and define S e as the total extinction cross 
section per cubic centimeter. From equations (34) 
and (43), it is then found, by adding up or integrating 
the cross sections of all bubbles contained in one 
cubic centimeter, 




4rrR l n(R) ( ~ 


dR 


(56) 


- 1 + 5 2 


Bubbles of near-resonant radius will make a large 
contribution to S c . If n(R) does not change rapidly 
for radii near resonance, the integral over the reso¬ 
nance peak in equation (56) may readily be evaluated. 

This procedure gives the correct value for S e pro¬ 
vided that absorption by bubbles far from resonance 
can be neglected. Even if the density of bubbles near 
resonance is comparable with the bubble density at 
other radii, resonant bubbles will probably make the 
major contribution to S e , since a e is unquestionably 
much greater for resonant bubbles than for those of 
other sizes. However, according to what has been 
said in Section 28.1.2 about the gradual shrinkage of 
bubbles, a large number of very small bubbles are 
likely to be present which may contribute apprecia- 





470 


ACOUSTIC THEORY OF BUBBLES 


bly to the total extinction cross section. Since <j, for 
bubbles of sizes far from resonance size is propor¬ 
tional to 8, and since the value of this damping con¬ 
stant is unknown for nonresonant bubbles, it is not 
possible to state the conditions under which non¬ 
resonant absorption may become important. In 
practical applications it is customary to assume that 
bubbles near resonance provide the dominant source 
of attenuation in wakes; as shown in Chapter 34, 
this assumption appears to lead to agreement with 
experimental results, and is probably correct at 
supersonic frequencies for the bubble distributions 
occurring in wakes. 

Then, in order to compute the value of S e resulting 
from bubbles near resonant size, n(R), v(R,f), and 
8(R,f) in equation (56) may be taken outside the 
integral and given their values for R equal to the 
resonant radius R r . Thus equation (56) is trans¬ 
formed into 


S 


e 


47t R*n(R r )8 r 

Vr 



(57) 


As the radius of the resonant bubbles struck by a 
sound beam of the frequency / is R r , then according 
to equation (23) 

(2x/ r ) 2 = 3 ^°, (58) 

ph r 


and from equations (23) and (58) 

fr Rr 

f - R ’ 

and according to equation (49) 


(59) 


Q f R ’ 


dq = 


(60) 


By substituting equation (60) into equation (57), S e 
can be expressed by an integration over the variable 
q. Only the values near the peak (near to q equals 1) 
make a considerable contribution to the value of the 
integral. Therefore, the transformations (49) and (50) 
may be used, and from equations (57) and (60) it 
follows that 

8. = ****& r • _JL_. (61) 

yjr «/—co 4 q~ T 8 r 

The integral has been extended to infinity. This 
simplification can be made because on this approxi¬ 
mation the contributions which are not very near to 
the peak can be disregarded. Evaluating the integral 
in equation (61) gives 


r+” dq t r 

J- » 4 q 2 + 8 2 r = 28 r 

and from equations (61) and (62) 

„ 2 T"R 3 MR r ) 

S e — 

Vr 

Let now u(R)dR denote the total volume of air con¬ 
tributed by the bubbles with radii between R and 
R + dR in 1 cu cm of the air-water mixture. Hence, 

u(R) =jR 3 n(R), (64) 


(62) 


(63) 


and from equations (63) and (64) 


& = 


3irU(R r ) 

2t] r 


(65) 


The quantity ?? r , according to equation (23), has the 
value 1.36 X 10~ 2 in sea water at 60 F and at atmos¬ 
pheric pressure. Hence, 

S e = 346.5 u(R r ). (66) 


In computing the attenuation for a region containing 
bubbles of many sizes, the equations derived at the 
beginning of this section may be applied directly. 
It is necessary only to replace the factor n<r e in equa¬ 
tion (53) by S e , taken from equation (66). If this 
substitution is made, the coefficient of attenuation is 


K e = 396.8 X 346.5 X u(R r ) 
I\e = 1.4 X 10 5 u(R r ). 


(67) 


This expression is the generalization of equation (55) 
for bubbles with a wide dispersion in size. It will be 
used in Chapter 34 to compute the amount of air in 
wakes from the observed attenuation coefficients. 


28.3.3 Scattering 

In accordance with the picture for propagation of 
sound through a region containing bubbles, as pre¬ 
sented in Section 28.3.1, the basic equation for scat¬ 
tered sound is very simple. The scattered sound in¬ 
tensity from a region is, on the average, simply the 
sum of the intensities of the waves scattered by each 
bubble. For a single bubble, the intensity at a dis¬ 
tance r is given by the equation 

I a = . , h > (68) 

4xr- 

where I 0 is the intensity of the incident sound at the 
bubble. This equation may be found from equations 










SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 


471 


(3), (5), and (6); more simply, it may be written down 
directly, since by definition <r s I n is the rate at which 
sound is scattered by a single bubble, and since the 
energy is spread out uniformly in all directions, at the 
distance r it is spread out uniformly over an area 
47rr 2 . In a small region of volume dV, the number of 
bubbles is nd V, where n is the number of bubbles per 
cubic centimeter. 

Equation (68) must be modified to allow for the 
tact that the scattered sound will be attenuated on 
its way from the region to a distance r away. Over 
long distances various sources of attenuation must be 
considered, such as absorption in the water, scat¬ 
tering by temperature irregularities, and so forth. 
Over short distances, most of these effects may be 
neglected, and the transmission loss taken from equa¬ 
tion (52). The basic equation for the scattered sound 
measured a distance i\ from the region dV then be¬ 
comes 


dl . = 


n<r,dV 

47r/'i 


Ie~ c 


Jl n(r)dr 


(69) 


where / is the intensity at the region dV. If sound 
from different directions is incident on the region, / 
must be averaged over all directions for use in equa¬ 
tion (69). 

Computing the scattered sound intensity from 
equation (69) is a much more complicated problem 
than computing the total sound attenuation from 
equation (51). In the latter case, equation (51) could 
be integrated along a single sound ray, yielding equa¬ 
tion (52) directly. The basic difficulty in solving equa¬ 
tion (59) is that the sound intensity I at the volume 
element dV includes sound scattered in turn from 
other regions. To consider multiple scattered sound 
of this type is rather complicated, and leads to inte¬ 
gral equations which in general cannot be solved 
exactly. Methods for treating this problem have been 
extensively explored in astrophysical literature. 10 ' 11 
The problem of multiple scattering in wakes could 
probably be studied with success by methods de¬ 
veloped for the corresponding optical problem. 12 

Fortunately, bubbles absorb much more sound 
than they scatter. From equation (46) and Figure 2 
it is evident that the ratio a s /cr e for resonant bubbles 
is less than 1 to 10 for frequencies above 15 kc. For 
this reason, sound scattered several times from 
resonant bubbles has usually traveled so far that it 
is very weak. Multiple scattering will therefore be 
neglected in all subsequent discussions. In simple 
cases, the error resulting from this approximation 


will be less than half a decibel at frequencies above 
15 kc. Even at 5 kc, the error will usually be less than 
1 db. For scattering by nonresonant bubbles, multiple 
scatterings cannot be neglected unless <r e is much 
greater than o- s . 

Even with this approximation, the computation of 
I a from equation (69) is not simple. The quantity I 
now becomes the sound intensity incident on the 
region containing bubbles, and attenuated by its 
passage through part of the region. However, to 
compute I s at any one point the sound arriving from 
all parts of the screen must be computed; the total 
scattered sound must be evaluated by summing up 
the contributions arriving from all different direc¬ 
tions. In any practical situation, the directivity of 
the receiving hydrophone must also be taken into 
account in order to find the electrical signal received 
in the measuring equipment. A detailed considera¬ 
tion of these problems in cases of practical impor¬ 
tance is given in Chapter 34. 

To give insight into fundamental features of the 
scattering problem, it is desirable to eliminate these 
geometrical complications as far as possible. Equa¬ 
tion (69) is here applied to scattering from a bubble 
screen, that is, from a layer of aerated water bounded 
by two parallel planes a distance w apart. Instead of 
integrating over all directions, we shall compute 
simply the scattered sound reaching the point P from 
all directions within a small cone of solid angle dQ; 
the quantity dQ is simply the area of a cross section 
of the cone divided by the distance r 2 from P to the 
cross section. The geometry of this situation is shown 
in Figure 3. 

Let I (0) be the intensity of sound incident on the 
screen; the incident sound is assumed to be a plane 
wave, whose rays are inclined at an angle i with a 
line perpendicular to the boundary of the screen. 
Within the screen the intensity falls off exponen¬ 
tially; since the path length dr is equal to sec idx, 
equation (52) gives for the incident sound at a dis¬ 
tance x inside the screen 


I(x) = 1(0) exp 


[ 


— a, sec 



As Figure 3 shows, the scattered sound which we are 
considering makes an angle e with a line perpendicular 
to the boundary of the screen. Thus in equation (69) 
the length dr along the path of the scattered sound is 
sec edx. Thus we find for the sound scattered from a 
small element of volume dV, at a distance ri from the 
point P 




472 


ACOUSTIC THEORY OF BUBBLES 



Figure 3. Scattering from a bubble screen. 


dJi= n(^jdV m 


4 7T/'“ 


exp 


rr,. (sec i + sec e) n(x)dx 


I" 

Jo 


(70) 


For the volume element dV within the cone, we have 
dV = r^dUdr; 


since dr i is simply sec tdx as before, equation (70) be¬ 
comes 

n(x)(T s sec edttdx 
dls= -:- 7(0) 


exp 



o-,(sec i + sec e) I n(x)dx 

Jo 


(71) 


This equation may be integrated over x from 0 to w, 
yielding 


dl s = 


a, (19. 


a Air cos i + cos t 


1 - exp 


o>(sec i + sec e) I n{x)dx 


f- 


(72) 


It is interesting to note that dl s in equation (72) is 
independent of the distance from the screen to the 
point P where the scattered sound intensity is 
measured. This apparent contradiction is resolved 
when it is realized that with increasing distance a 
larger area of the screen is intercepted within the 
solid angle d9l. 


Equation (72) has two important limiting cases. 
When the transmission loss across the screen is large, 
the second term in the brackets is negligibly small, 
and 


dJ s = 


<x s d9 

(t c 4tt 


cos i 

cos i -f- cos e 


(73) 


It may be noted that when e equals i, as is the case for 
backward scattered sound, cos e equals cos i ; if also 
equation (46) is used for u s o>, equation (73) yields 


dh 


r, dtt 
8 Sir 


(74) 


On the other hand, when the transmission loss 
across the wake is small, it is possible to use the ap¬ 
proximate relationship 

e~“= 1 - a, 


yielding 


dl s 



(75) 


In terms of the average density n introduced in the 
previous section, equation (75) becomes 

dtt 

dl s = (t s — sec e wn- (76) 

4ir 


Thus when the transmission loss across the wake is 
small, dl s is proportional to cr s and n. But when the 
transmission loss is great, the scattered sound reaches 
a constant value, given by equation (73), and is in¬ 
sensitive to changes in n or w. 

When bubbles of different sizes are present, equa¬ 
tion (69) for dl s may still be used, provided that na e 
is replaced by S e , the total extinction cross section 
per cubic centimeter, and na, is replaced by S s , the 
total scattering cross section per cubic centimeter. 
The quantity S e is discussed in the preceding section; 
equation (65) gives the relationship between S e and 
u(R r ), the bubble density at resonance. A similar 
analysis, considering bubbles only of near-resonant 
size, leads to the following equation for the total 
scattering cross section per cubic centimeter: 


3iru(R r ) 
2 8 r 


(77) 


The consideration of only those bubbles near the 
resonant size is usually legitimate even if absorption 
by nonresonant bubbles is appreciable. Since <r s , the 
scattering cross section of a single bubble, does not 
depend on the damping constant 8 for nonresonant 
bubbles, it is possible to evaluate precisely the con¬ 
tribution of bubbles of all sizes. For a single bubble 



















SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 


473 


smaller than resonant size, <r s falls off as the fourth 
power of the wavelength; hence such small bubbles 
are not likely to contribute much to S, unless present 
in very large numbers. Bubbles larger than resonant 
size have a scattering cross section about four times 
their geometrical cross section, but are not likely to 
be present in greater abundance than smaller bubbles. 
Thus equation (77) should be valid in a wide range 
of circumstances. 

Since the ratio of S a /S e is equal to the ratio of 
a,/<r e at resonance, equation (74) is still valid when 
the transmission loss across the screen is large; thus 
the scattered sound in this case is just the same as if 
all bubbles were of resonant size. When the trans¬ 
mission loss across the screen is small, however, equa¬ 
tion (76) must be used, with S s substituted in place 
of nov 


28.3.4 Reflection and Refraction 


The presence of bubbles changes the velocity of 
sound. If the bubble density is sufficiently great, this 
effect may become practically important, leading to 
reflection and refraction of the sound beam. Since in 
ship wakes the number of bubbles present per cubic 
centimeter is usually not sufficiently great to change 
the sound velocity very greatly, these effects are not 
discussed in great detail here. The methods of analysis 
required to deal with this case are briefly sketched, 
and the results stated. 

The sound velocity is defined by equations (6), 
(18), and (26) in Chapter 2 as 

c 2 = ^ , (78) 

dp 


where p and p are the pressure and density respec¬ 
tively of the bubble mixture. If only a volume V of 
the mixture is considered, equation (78) may be 
written in the form 


F — 
_df 

8 V 

P dt 


(79) 


using the relation pdV -f Vdp = 0. The quantities 
dp/dt and d V / dt maybe evaluated from the equations 
in Sections 28.1 and 28.2 yielding the basic equation 


4 = 1-/; 


n(R)RdR 


f 

h - 1 


+ is 


(80) 


where c 0 is the sound velocity when no bubbles are 
present, and n(R) is the number of bubbles per cubic 
centimeter with radii between R and RdR. The inte¬ 
gral in equation (80) extends over all bubble sizes. 
It is assumed that all bubbles present have a radius 
much smaller than the wavelength, and that the 
average distance between bubbles is larger than their 
radius. If these two assumptions are not fulfilled, the 
theory on the preceding pages breaks down. The de¬ 
tails of the derivation of this equation are given in 
references 3 and 4. 

It may be noted that equation (80) is valid only 
when the density of the liquid-bubble mixture is 
substantially the same as that of the liquid. Results 
are given which may be used for any density of 
bubbles, provided that the bubbles are all much too 
small to resonate, but much too large for surface 
tension to become important. 13 

For frequencies far from resonance, the imaginary 
term in equation (80) may be neglected. For fre¬ 
quencies below resonance, this leads to the equation 


= 1 + 


3 a 

2 

Vr 


(81) 


where u is the total volume of air present as bubbles 
in 1 cu cm of the liquid-bubble mixture. Thus u is de¬ 
fined by the equation 


= f— ft 3 n(i?)d/?. 

«/ 3 


(82) 


The quantity rj r in equation (81) is the ratio of the 
bubble circumference 2irR to the wavelength X at 
resonance, as defined in equation (1). Equation (81) 
is valid only for bubbles which are sufficiently large 
that surface tension effects can be neglected; more¬ 
over, if the expansion and contraction of the bubble 
are adiabatic rather than isothermal, the last term in 
equation (81) must be multiplied by the ratio of the 
specific heats for the gas in the bubble. It is interest¬ 
ing to note that, subject to these limitations, equa¬ 
tion (81) is independent of the bubble radius. Even 
if u is as low as 10 -4 parts of air at atmospheric pres¬ 
sure to one part of water, c/c 0 is 0.62. 

When the bubbles are all greater than the resonant 
size, the sound velocity is increased by the presence 
of the bubbles, and the relation corresponding to 
equation (81) is 

4 = l-^f§ z u(R)dR, (83) 

C T] r J h 

where u(R)dR, defined in equation (64), is the volume 
of the bubbles present in 1 cu cm of liquid-bubble 




474 


ACOUSTIC THEORY OF BUBBLES 


mixture with radii between R and R + dR. Equation 
(83) has the surprising implication that when u(R) is 
sufficiently great, Cq/c 2 becomes negative, the sound 
velocity becomes purely imaginary on this approxi¬ 
mation, and the attenuation becomes very large. 
Under these circumstances the imaginary term in 
equation (80) which was neglected in equation (83), 
determines the wave velocity and the wavelength. 
For the case of all bubbles with twice the resonant 
radius R r , the critical value of u at which c becomes 
infinite is 2 X 10 -4 , corresponding to a distance be¬ 
tween bubbles of roughly thirty times the bubble 
radius. 

When resonant bubbles are present, the imagi¬ 
nary part of the sound velocity becomes important. 
If an integration is carried out only over bubbles 
close to resonance, and if u(R) is not changing rapidly 
with R in this region, the real part of the integral in 
equation (80) is small and may be neglected, yielding 


Cq 3iriR r u(R r ) 

t 2i] r 


(84) 


This imaginary part of the sound velocity leads to an 
exponential decay of sound intensity with distance x, 
since the sound intensity falls off as e -2,r,/r c . If the 
second term on the right-hand side of equation (84) 
is small, as it is in most practical cases, the resulting 
attenuation is exactly the same as was found in equa¬ 
tions (53) and (67) in Section 28.3.2. 

In a region containing bubbles, with any assumed 
distribution of sizes, and having a sharp boundary, 
sound incident on this region from bubble-free water 
will be reflected at the sharp discontinuity. The anal¬ 
ysis for this situation is given in Section 2.6.2 where 
it is shown that the ratio of the amplitude of the 
reflected and incident waves is given by the equation 


A" Ci — Co (cos e/cos t) 
•do Ci + Co (cos e/cos t) 


(85) 


This is essentially equation (119) of Chapter 2, with 
Pi set equal to p' and subscripts 0 used for the incident 
sound wave. The quantity Co is the sound velocity in 
the bubble-free medium, while Ci is the corresponding 
quantity across the boundary, where bubbles are 
present. The energy reflection coefficient y r is simply 
the square of A"/A 0 . The angles i and 6 are the angles 
which the incident and refracted sound make with a 
line perpendicular to the boundary. The ratio of cos e 
to cos l may be found from Snell’s law, yielding 


COS 2 € 
COS 2 L 


1 + tan 


■‘(■-I) 


In most cases of practical importance, Ci is nearly 
equal to c 0 . Thus, cos e is essentially equal to cos i. 
By writing equation (81) in the form 

c l . , , 

—y = 1 + 0 ) 

C 

the energy reflection coefficient y e found by squaring 
A"/A 0 in equation (85) becomes 

7«: = 77, > (86) 

16 

as long as b is much smaller than 1. This equation 
may also be used when b is complex but less than 1, 
provided that the absolute value of b is used. When 
b is comparable to or larger than 1, the formulas be¬ 
come considerably more complicated. 4 

28.3.5 Observed Acoustic Effects of 
Bubbles 

The effect of a known distribution of bubbles on 
the propagation of sound through water has been 
investigated in the laboratory at frequencies from 
10 to 35 kc. 8 The method used for producing a screen 
of bubbles all of the same size has already been de¬ 
scribed in Section 28.2.1. Special measurements were 
made to determine the number of bubbles per cubic 
centimeter at various points in the bubble screen. 

The bubble screen was about 17 in. ong. Its thick¬ 
ness varied with the bubble radius; for bubbles 0.034 
cm in radius, corresponding to a resonant frequency 
of 10 kc, the thickness was about 3 in., while for 
bubbles 0.020 cm in radius, corresponding to 17 kc, 
the thickness was more nearly 5 in. In continuous 
flow, about 1 cu cm of air per second was fed into the 
screen, resulting in a total density u of about 10 -4 
parts of air per part of water. When a bubble pulse 
was formed by turning on the stream of bubbles for 
1 sec, however, the bubble densities at the level of 
the acoustic instruments were much less than this, 
ranging between 10 -6 and 10~ 7 . 

The acoustic measurements with the bubble pulse 
consisted in measuring the sound reflected from and 
transmitted through the screen at a fixed frequency 
as a function of time since the beginning of the pulse. 
The transmission loss was measured by reading the 
sound level in a hydrophone placed on the far side 
of the bubble screen from the projector. The reflected 
sound was measured by a hydrophone placed on the 
same side of the bubble screen as the projector, but 
separated from the projector by several baffles. The 





SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 


475 


RADII IN CM OF BUBBLES IN SCREEN 
>0.070 0.040 0.030 0.025 0.020 0.017 aOI5 0.014 0013 &0I2 0.011 



RADII IN CM OF 8UBBLES IN SCREEN 
>0.070 0.040 0.030 0.025 0.020 0X117 0.015 0.014 0.013 0.012 0.011 



Figure 4. Acoustic data taken with bubble pulse 
screen at 20 kc. 


hydrophone was placed symmetrically with the pro¬ 
jector, so that the sound reflected specularly from 
the screen could reach the hydrophone. However, 
scattered sound could also reach the reflection hydro¬ 
phone, and presumably contributed to the so-called 
reflected sound which was measured. 

As the resonant bubbles passed by the level of the 
acoustic measuring instruments, the transmitted 
sound intensity showed a sharp dip. The reflected, or 
scattered, sound showed a very much broader maxi¬ 
mum, in agreement with the constant scattered sound 
intensity predicted by equation (73) whenever the 
transmission loss through the region is appreciable. 
However, the reflected sound showed much greater 
fluctuations than the transmitted sound. 

Sample records of transmission through and re¬ 
flection from bubble screens are reproduced in Figure 
4. The curves show the output of the transmission 
and reflection hydrophones as a function of time 
elapsed after a 1-sec pulse of bubbles was formed 6 ft 
below the acoustic equipment. Also shown are the 
radii of the bubbles arriving at each time. These 
radii were measured directly by visual means. The 
upper diagram in Figure 4 shows three transmission 
runs at 20 kc, superposed on each other. The radius 
of the resonance peak in this diagram agrees well with 
the theoretical value of 0.017 cm found from equa- 



0 0.01 0.02 0.03 0.04 

BUBBLE RADIUS ON ACOUSTIC AXIS IN CM 


CD 

O 


UJ 

LU 

01 

O 

c n 

x 

o 

3 

O 

01 

X 


SOUND FREQUENCY IN KC FOR RESONANCE 

10 



BUBBLE RADIUS ON ACOUSTIC AXIS IN CM 



0 0.01 0.02 0.03 0.04 

BUBBLE RADIUS ON ACOUSTIC AXIS IN CM 


Figure 5. Resonant attenuation and reflection with 
bubble pulse screen. 


tion (18). The lower diagram shows a reflection run 
at 20 kc. The measured reflection coefficient is the 
difference in level between the incident sound at the 
bubble screen and the sound measured with the re¬ 
flection hydrophone, placed 2.5 ft from the center 
of the screen. 

Each set of observations was repeated at least 
three times at each of several frequencies. Before and 
after each group of acoustic measurements, detailed 





































































476 


ACOUSTIC THEORY OF BUBBLES 



£ 0.015 0.020 0.025 0.030 0.035 


BUBBLE RADIUS IN CM ON ACOUSTIC AXIS 



8UBBLE RADIUS IN CM ON ACOUSTIC AXIS 


-Upper and lower limits of experimental data 

- Estimated average intensities 

Figure 6. Scattering and reflection bubble pulse 
screen. 

observations were made of the number of bubbles of 
different sizes in the screen, since the operation of the 
microdispersers producing the bubbles tended to be 
somewhat erratic. If the physical measurements on 
the number of bubbles of different sizes in the screen 
did not give the same results before and after the 
acoustic measurements, the acoustic data were dis¬ 
carded. 

The results of the acoustic measurements on bubble 
pulses showed moderate agreement with theoretical 
predictions. Figure 5 illustrates typical results ob¬ 
tained for resonant bubbles. The upper diagram 
shows the total number of bubbles per cubic centi¬ 
meter at the level of the transducers at the time when 
bubbles of each radius reach that level. Since the 
spread of bubble radii at each time was small com¬ 
pared to the width of the resonance peak for a single 
bubble, all the bubbles at any one time may be as¬ 
sumed to be of the same size. In the middle diagram, 
the continuous curve shows the predicted attenuation 
through the screen, found by substituting in equation 
(53) the following quantities: the bubble density 
taken from the upper diagram; the measured thick¬ 


ness of the screen; and the value of <r e found from 
equation (43) with/equal to f r , and with values of 8 r 
taken from Figure 2. The average observed trans¬ 
mission losses at each frequency are shown by circles, 
with vertical lines showing the spread of the observa¬ 
tions. These experimental points are essentially the 
maximum difference in sound level produced by the 
passage of the bubbles; in the middle diagram of 
Figure 5, for example, the observed resonant trans¬ 
mission loss at 20 kc is about 14 db. 

The lower diagram in Figure 5 shows the intensity 
of the reflected or scattered sound for resonant 
bubbles. To compute the reflection to be expected 
from resonant bubbles, the specular reflection was 
first found from equation (86), with b evaluated for 
bubbles all of resonant size. To this was then added 
the scattering to be expected; this scattered sound 
was found from equation (76), since for resonant 
bubbles the transmission loss through the screen was 
always great enough to make this equation appli¬ 
cable. The value of rjr at resonance was taken from 
equation (23), while values of <5 r at resonance were 
again found from Figure 2. In the computation of this 
scattered sound account must be taken of the size of 
the screen and its distance from the sound projector 
and hydrophone. The solid curve in Figure 5 shows 
the theoretical predictions; at 10 kc, specular reflec¬ 
tion is most important, while at 30 kc, scattered 
sound is dominant. 

A similar comparison between theory and observa¬ 
tion may be made for nonresonant bubbles. The 
transmission loss measurements yield nothing further 
of interest, since the width of the observed resonance 
curve has already been used to find values of 5 r . For 
bubbles whose size is so far from resonant size that 
the transmission loss is small, the scattering may be 
predicted from equation (75), suitably modified to 
take into account the geometry of the situation. The 
value of a s to be used may be taken from equation 
(34). Specular reflection from nonresonant bubbles is 
negligible. Plots of the observed data are shown in 
Figure 6, where the crosses represent the computed 
values for nonresonant scattering. The spread of the 
observations is indicated by the dashed lines, with 
the solid line showing the estimated average in¬ 
tensities. The circles represent the predicted scatter¬ 
ing and reflection from resonant bubbles, already dis¬ 
cussed. The dotted linegives the sound levelmeasured 
at the reflection hydrophone when no bubbles were 
present. 

It is evident that the agreement between theory 

























SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 


477 


and observation shown in Figures 5 and G is not bad. 
Other runs show about the same agreement, with oc¬ 
casional observed transmission losses as low as half 
or as great as twice the predicted value, and with 
occasional observed reflection coefficients as much as 
6 db outside the spread of the observational data. 
These discrepancies, which are apparently in random 
directions, may be the result of irregularities in the 
bubble-producing devices. It is worth noting that the 
predicted scattering from nonresonant bubbles should 
be quite reliable, since the theoretical values are in¬ 
dependent of the damping constant. Hence it may be 
concluded that the agreement of observations with 
theory is within the observational error, and justifies 
the practical use of the equations developed in this 
chapter. 

Measurements on continuous-flow bubble screens 
have also been described; 7 they showed relatively 
poor agreement with the theoretical predictions. The 
observed transmission losses rarely exceeded 25 db, 
while the predicted transmission losses ranged be¬ 
tween 50 and 200 db. It is doubtful whether such 
great transmission losses could be observed, since 
sound diffracted around the screen would be expected 
to become important. In addition, in the continuous- 
flow screen the smaller bubbles extended over a wider 
region than the larger ones. At the lower supersonic 
frequencies this halo of small bubbles would not 


absorb sound, but would reduce the sound velocity, 
thus tending to bend the sound rays around the 
screen. 

Furthermore, the predicted reflection coefficients 
for the continuous-flow screen were some 5 to 15 db 
greater than the observed values. The high specular 
reflection predicted from theory for these continuous- 
flow screens would presumably be reduced to a value 
closer to the observed results if account were taken 
of the absence of sharp boundaries. In view of the 
many complexities entering into the explanation of 
these measurements on the continuous-flow screen, 
these discrepancies with theory may be disregarded. 

An important theoretical question which is not 
answered by these experiments is the absorption pro¬ 
duced by bubbles far from resonance. This non¬ 
resonant absorption depends on the variations of <5 
with bubble radius and sound frequency. Since the 
values of 8 r are unexplained, the predictions of theory 
as regards values of 8 under other conditions are of 
little use. The bubble pulse measurements show that 
the absorption by nonresonant bubbles is usually less 
than about 5 per cent of the absorption by resonant 
bubbles. It is not impossible that for some bubble 
distributions present in wakes nonresonant absorption 
might be practically important. Further observations 
under controlled conditions would be required to 
cast light on this point. 



Chapter 29 


VELOCITY AND TEMPERATURE STRUCTURE 


T he water in the wake of a ship is usually in mo¬ 
tion relative to the surrounding water. In addi¬ 
tion, the temperature of the water at different points 
in the wake is sometimes characteristically different 
from the temperatures found outside the wake. The 
variations of temperature and velocity are important- 
physical properties of wakes, and might be expected 
to account at least in part for the acoustic effects ob¬ 
served; furthermore, a study of these physical char¬ 
acteristics is of independent military interest. Even if 
air bubbles are responsible for all the observed 
acoustic effects of wakes, any theory of the origin and 
persistence of bubbles must be consistent with known 
facts about the velocity and temperature structure. 

The present chapter summarizes the fragmentary 
evidence which is available on these two subjects. 
Sections 29.1 and 29.2 discuss the available data on 
the velocity and temperature, respectively. In Section 
29.3 the resulting acoustic effects are examined. It is 
shown that scattering from turbulent but wake-free 
water is negligible; scattering of sound by water with 
an irregular temperature distribution may some¬ 
times be appreciable, but cannot explain the large 
acoustic effects observed. Thus, velocity and tem¬ 
perature structure alone cannot account for the ob¬ 
served acoustic properties of wakes. 

29.1 VELOCITY STRUCTURE OF WAKES 

The simplest wake is that produced by the flow of a 
fluid past a thin plate parallel to the stream. In this 
case the plate affects the flow only in a narrow region 
close to the plate, known as the boundary layer, 
where the fluid is slowed down. This effect is shown 
in Figure 1, where the magnitude of the velocity at 
various points is shown by arrows; for simplicity, 
only the upper half of the flow pattern is shown. Far 
behind the plate the velocity distribution still shows 
the effect of passing by the plate, since the fluid which 
passed through the boundary layer will be moving 
less rapidly than the rest of the stream. The arrows in 


Figure 1 represent the average velocities of the fluid 
relative to the plate. Thus these results are applicable 
directly to the reciprocal situation, when the thin 
plate (or ship’s hull) is moved through still water. In 
this situation the water in the wake is left moving in 
the same direction as the plate. It may be noted that 
in most cases, the flow in the boundary laj'er becomes 
turbulent, in which case the flow in the wake will also 
be turbulent. 



UNDISTURBED BOUNDARY WAKE 

FLOW IAYER 

Figure 1. Velocity structure. 


In addition to the wake produced in this way by 
passage of a ship through water, there is also the 
effect produced by the screws. To move the ship for¬ 
ward, the screws exert a forward force on the ship 
which is somewhat greater than the frictional force 
produced by the flow of water past the hull; the dif¬ 
ference is just equal to the retarding force due to wave 
action and air resistance. For a submerged submarine, 
however, the propulsive force is just equal to the fric¬ 
tional force produced by the flow of the water around 
the hull. To produce this propulsive force on the sur¬ 
face ship or submarine, the screws exert an equal and 
opposite force on the water, which is forced back¬ 
ward. As a result, the water passing through and 
around the screws moves in a direction opposite to 
that of the vessel. The flow of water produced by 
ship screws has already been discussed in detail in 
Section 27.1.1 in connection with the formation of air 
bubbles. 

Thus, close to a ship the wake is made of several 
component parts: one or more screw wakes, usually 
called “slipstreams,” moving away from the ship as a 
result of screw action; and the hull wake following 
the ship as a result of frictional force at the surface of 


478 






















TEMPERATURE STRUCTURE OF WAKES 


479 


the hull. The backward momentum of the slipstream 
is nearly canceled out by the forward momentum 
of the hull wake, except at surface ship speeds so 
high that wave resistance becomes the most im¬ 
portant retarding force on the ship. In the wake of a 
submerged submarine this cancellation is exact. 

At moderately close distances astern, probably 
much less than a ship length, these different streams 
become intermingled and confused, giving rise to a 
turbulent mass of water in which velocities in almost 
any direction are equally likely. Over a small distance 
called the patch size, the velocity at any one time is 
reasonably constant, but the velocity at any point 
fluctuates rapidly. Information on turbulent motion 
is rather incomplete and no velocity measurements 
are available in surface ship or submarine wakes. As 
noted already in Section 27.2, not much is known 
about the magnitude of the turbulent velocities, the 
average patch size of the turbulent elements, or the 
rate at which the turbulence gradually dies away. 

29.2 TEMPERATURE STRUCTURE OF 
WAKES 

The water temperature at different points in a 
wake has been the subject of more study than the 
water velocity. This is partly because small tem¬ 
perature differences can be measured much more 
readily at sea than small fluid velocities. By the use 
of sensitive thermopiles fastened to a surface vessel, 
temperature fluctuations as small as 0.01 F may be 
readily recorded. Data obtained with this technique 
at the U. S. Navy Radio and Sound Laboratory 1 and 
elsewhere show that the presence or absence of ob¬ 
servable temperature structure in wakes depends on 
the presence of vertical temperature gradients in the 
sea before the passage of the ship. 

29.2.1 Constant Temperature in 
Surface Layer 

When a ship is passing through water all of the 
same temperature, such as is commonly found in the 
top 50 ft of the ocean, especially during winter 
months, no thermal structure in the wake can be 
observed. Repeated wake crossings under these con¬ 
ditions have failed to show any trace of temperature 
structure. In such isothermal water, temperature 
structure could be produced only by the heating ac¬ 
tion resulting from the passage of the ship. Such 
heating can readily be shown to be negligible. 


To consider an extreme case, suppose a ship at 
30 knots is exerting 30,000 hp, and suppose that all 
this energy goes into heating a wake with a cross sec¬ 
tion 20 ft square. The increase of temperature result¬ 
ing in this extreme case is 0.015 F. In most practical 
cases, the temperature change will be very much 
smaller. Although small patches of water might be 
appreciably warmed by water discharged from cool¬ 
ing systems, by dissipation of energy in intense 
vortices, or by similar processes, most of the wake 
behind a ship in isothermal water will have a tem¬ 
perature which is practically the same as that of the 
surrounding ocean. 

29.2.2 Temperature Gradient in 
Surface Layer 

When a vertical temperature gradient is observed 
in the top 20 ft of the ocean, the passage of a ship 
disturbs the temperature structure and gives rise to a 
measurable temperature structure in the wake. The 
thermopiles used in research on this subject have had 
slow response times, requiring 1 or 2 sec for 80 per 
cent response; since the surface vessels used in the 
work were under way at 3 knots or more, changes of 
temperature over regions less than a few feet in length 
could not be detected. 

The most detailed and quantitative work 1 was 
carried out with four thermopiles attached to a long 
pipe mounted vertically on the bow of a small cabin 
cruiser; the thermopiles were at depths of 4, 6, 8, and 
10 ft below the surface. In each thermopile, one set of 
junctions was thermally exposed to the sea water; 
the other set was thermally insulated and remained 
at the average temperature of the surrounding water, 
averaged over a period of minutes. The output of 
each thermopile was measured with a self-balancing 
potentiometer; since these instruments required some 
7 sec to reduce an unbalance to zero, these quanti¬ 
tative measurements recorded only the large-scale 
features of the wake thermal structure. 

Results obtained with this technique are shown in 
Figure 2, obtained in successive crossings of a 
destroyer wake 8 and 15 minutes old. Accompanying 
bathythermograph records are also shown. It is 
evident that the fresh wake consists of warmer water 
at the two sides, with cooler water in the middle. This 
distribution probably results from descending cur¬ 
rents at the sides, and rising currents in the center; 
such currents coidd be produced by the rotation of 
the slipstreams from the two propellers. 



480 


VELOCITY AND TEMPERATURE STRUCTURE 


WAKE AGE 8 MINUTES 
LAUNCH ENTERING FROM WEST 



0 

h 10 

£ 20 
z 

i 30 

t 

UJ 

Q 40 
50 

BT AT 1216 


61 

62 F 

. 

T 

- 


- 


- 


' 





WAKE AGE 15 MINUTES 
LAUNCH ENTERING FROM EAST 



K 

1- 10 

4 5 

UJ 

UJ 

2 

“■ 20 

cc 

z> 

z 

CO 

6 2 

5 

EPTH 

Ol 

O 

u. 

O 

0 

8 o 


I 


t- 

CL 

50 

LJ 


10° 



61 F 

62 F 


T- 






BT AT 1316 


EAST SIDE WEST SIDE 

OF WAKE OF WAKE 


Figure 2. Horizontal temperature structure of a de¬ 
stroyer wake. 


The thermal structure found for other types of ship 
wakes is sometimes considerably different from that 
shown in Figure 2, with single peaks sometimes re¬ 
placing the double peaks. In general, however, when¬ 
ever the thermopiles were at the depth of a marked 
negative gradient — 0.5 degree in 10 ft — as shown 
on a bathythermograph record outside the wake, the 
wake near the surface was colder than the surround¬ 
ing water at the same depth. When the gradient is 
marked no such general rule may be made. It is 
interesting to note, however, that thermal wake 
signals have been readily detected when the gradient 
outside the wake was almost too weak to be noticed 
on a bathythermograph record — about 0.2 degree 
in 20 ft. 

Measurements have also been made on the thermal 
properties of the wake behind a submarine at peri¬ 
scope depth, with a moderate negative gradient pres¬ 
ent in the surface layer. It was found that effects ap¬ 
peared even at the surface, where the water behind 
the submarine was found to be a few tenths of a de¬ 
gree cooler than the surrounding water outside the 


wake. The reason for this rise of the submarine’s 
thermal wake to the surface is not known. 

The persistence of these thermal effects is some¬ 
times quite marked. Identifiable thermal signals have 
been obtained in crossing wakes an hour or more 
after these were laid. Not all wakes exhibit identi¬ 
fiable thermal effects for such a long period, even if 
the gradient is marked. The limiting factors are the 
decay of the thermal structure of the wake and the 
background of thermal irregularities present outside 
the wake. It is sometimes difficult to distinguish the 
thermal change found in crossing a wake from those 
frequently found in sailing through wake-free water. 
The thermal irregularities in wake-free water also 
tend to increase with increasing temperature gradi¬ 
ents; thus a very strong gradient is not necessarily 
the best for detecting a wake by its thermal proper¬ 
ties. 

As shown in the next section, the acoustic effect of 
thermal structure is greatest for temperature irregu¬ 
larities whose size is about equal to the wavelength 
of the sound being transmitted through the water. 
Thus, to compute the scattering of supersonic sound 
at 24 kc, information on the variation of temperature 
over regions about 3 in. long would be required. No 
such information is available, owing to the long time 
constants of the measuring methods discussed above. 
Temperature fluctuations over such small regions 
might be expected in a relatively fresh wake. How¬ 
ever, it would be surprising to find such a small-scale 
temperature structure in a wake more than a few 
minutes old. 

29.3 SCATTERING BY TEMPERATURE 
AND VELOCITY STRUCTURES 

Any region in which the velocity of sound varies 
with position will affect a sound wave passing through 
it. For example, theory predicts appreciable reflection 
from a surface separating two large bodies of water 
differing considerably in temperature. 2 If variations 
ol the microstructure of the ocean take place over 
distances not too great compared with the wave¬ 
length, an appreciable amount of sound will be scat¬ 
tered in various directions. Although the exact anal¬ 
ysis of these effects is complicated, certain results 
may be derived relatively simply. These results, given 
below, are sufficient to indicate the general magnitude 
of the scattering of sound by the temperature and 
velocity structure of wakes. 

Suppose that in some region S the velocity of sound 






























































SCATTERING 


481 


has some variable value c + Ac, while in the sur¬ 
rounding water the sound velocity has a constant 
value c. Suppose also that a plane sound wave, of in¬ 
tensity /o, and wavelength A, is progressing through 
the medium in the x direction. The intensity I a of the 
sound scattered from S may be different in different 
directions, but at long ranges will fall off as the in¬ 
verse square of the radial distance r from the center 
ol the region S. Since I s must be directly proportional 
to /o, 

= (1) 

where k is a constant. A more detailed discussion of 
this equation is given in Section 19.1 of this volume, 
describing in general the reflection, or scattering, of 
sound from objects or scattering regions in the sea. 
The target strength 7’ as usually defined is simply 
10 log k. 

The quantity k, which depends on the direction of 
the scattered sound under consideration, must be re¬ 
lated to the values of Ac, the sound velocity fluctua¬ 
tion, at different points in the region S. Only the 
energy scattered directly backward need be con¬ 
sidered here, since this corresponds to the situation 
of practical interest. It may also be assumed that 
the scattering is sufficiently small that the sound 
level at all points in S is practically equal to its value 
in the incident sound wave in the absence of scat¬ 
tering. This assumption tends to overestimate k ; if 
the scattering is large the sound level will decrease 
as the wave penetrates the region S, because energy 
is lost by scattering in the portion of the region S 
already passed through. 

Since the scattering is produced by the relative 
change in sound velocity, it is reasonable to assume 
(and, in fact, it can be shown) that the pressure dp s 
of the sound scattered from each volume element 
dxdydz in S is proportional to the value of Ac/c for 
each element. In adding up all the sound from differ¬ 
ent elements, the differences in phase must be con¬ 
sidered. Since sound must travel to the scattering 
element and then back along the x axis, the difference 
in phase between two elements separated by a dis¬ 
tance x along the x axis will be d-n-x/A. Thus to find 
the pressure of the scattered sound, Ac/c must be 
multiplied by cos (47r.r/\ + 2-irft), where / is the 
frequency of the sound, and integrated over the en¬ 
tire scattering region S. The scattered sound in¬ 
tensity is then proportional to the square of this 
integral. In this way it may be shown that the 


quantity k in equation (1) is given by the formula 

k = J'JJ’-f cos ( 4,r Y 2w f 1 ') dxdydz • (2) 


By writing 
'W.r 


( 47r.r \ 

»(- + 2,„)_ 


cos 47 t — cos 2irft 
X J 


— sin 47 t — sin 2irft , (3) 

X 

the integral in equation (2) becomes the sum of two 
integrals. Now square this sum, and average over the 
time t, using the relations 

cos 2 2tt ft — sin 2 2irft = \ (4) 

and cos 2irft sin 2irft — 0 , 

where the bars denote an average over the time t 
Then the quantity k, which measures the scattered 
sound intensity, becomes 

k ~ [I/J/t sin (t ) dxdydz ] (5) 

+ [7///7 cos (t)* :% *3 ! ' 

As pointed out above, the target strength of the 
scattering region is 10 log k. 

When the volume of the scattering region is small 
compared with the wavelength, the trigonometric 
functions in equation (5) are constant; since the 
sum of their squares is unity, 

k = ^(fffj dxdyds ) ■ (u) 

When Ac/c is constant throughout the region, this 
equation reduces to 



where T is the volume of the region. Equation (6) is 
the so-called Rayleigh scattering law, which predicts 
only a small amount of scattered sound. On the other 
hand, when c is constant over a region large compared 
with the wavelength, k is again small; as a result of 
the oscillation of the sine and cosine factors in equa¬ 
tion (5) each integral adds up to only a small value. 


29.3.1 Effect of Temperature 
Microstructure 

Equation (5) may be used to compute the sound 
scattered by a mass of water in which the tempera- 







482 


VELOCITY AND TEMPERATURE STRUCTURE 


ture varies rapidly from point to point. For sim¬ 
plicity, suppose that positive and negative values of 
c are equally likely — that is, that the average tem¬ 
perature of the water is just equal to the temperature 
outside the scattering medium. Although the distri¬ 
bution of temperature from point to point is a 
quantity which fluctuates at random, there is a 
certain patch size over which the temperature does 
not usually change appreciably. This is represented 
mathematically by means of the function p(f), which 
is defined by the expression 

, A c(x + f,y,z) A c(x,y,z) 

p(f)- - -’ (7) 

A c(x,y,z) 2 


than the patch size A, and the last term in the de¬ 
nominator may be neglected. Correlation coefficients 
of a form different from equation (10) do not gener¬ 
ally give a much greater value of k/V for a given 
patch size A. 

Numerical values may be substituted in equation 
(10). Fluctuations of 0.5 F with a patch size of 6 in. 
probably represent a rather extreme assumption. For 
this situation, k/V is about 3 X 10 -7 sq yd per cubic 
yard of volume. The volume scattering coefficient m 
discussed in Section 12.1 of this volume is related to k 
by the equation 


where the averaging is to be carried out in space, over 
all values of x, y, and z in the scattering region. While 
f is a displacement in the x direction in the expression 
(7) above, the displacement might also be extended 
in any other direction. The value of the function p(f) 
will depend both on the magnitude and on the direc¬ 
tion of f. If the displacement is zero, then p will equal 
unity. If the displacement is very large, the values of 
c at points separated by the distance r show no cor¬ 
relation with each other, and their product is alter¬ 
nately positive and negative, canceling out on the 
average; thus for large p approaches zero. The 
patch size is the value of f for which p becomes small, 
say less than about The function p is called a self- 
correlation coefficient. The temperature microstructure 
is described as isotropic if p(f) is independent of the 
direction along which f is taken. 

With some mathematical transformations, equa¬ 
tion (5) may be expressed in terms of p(f). For an 
isotropic medium, the resulting equation, which is 
equivalent to that given in a report by Columbia 
University Division of War Research [CUDWIl], 3 
is 

I671- 3 /A c\ : r°° sin (47rf/X) 

*—xKtDJ. (8) 

where V is the volume of the scattering region. As one 
fairly general type of possible correlation coefficient, 
it may be assumed 

p(f) = e- fM . (9) 

By substituting this expression in equation (8), and 
integrating, 

_k 1 7 

V = 8 ttT \ 

In actual practice the wavelength X is usually less 


7 


AcV 
c 


1 


(1 + X 2 /16t r 2 A 2 ) 2 


( 10 ) 


Thus m, in this case, is about 4 X 10 -7 per yard. If 
equal energy were scattered in all directions, m would 
be the fraction of energy scattered per yard of sound 
travel through the scattering medium. 

Evidently even these extreme assumptions give a 
very small scattering coefficient. Even if the scatter¬ 
ing volume is 10 yd thick, 30 yd across, and 100 yd 
long, corresponding to the wake in the path of a 
sound beam, k is about 1CU 3 yd, corresponding to an 
effective target strength of —30 db. The transmission 
loss through such a scattering region would be a very 
small fraction of a decibel. Temperature microstruc¬ 
ture cannot explain the strong echoes or the high 
transmission losses produced by wakes. 

29.3.2 Effect of Velocity 

Microstructure 

A separate analysis must be carried out for the case 
where the velocity of the water varies from place to 
place in the medium. This is a more complicated 
situation than the one in which the temperature 
changes, since the fluid velocity has a direction as 
well as a magnitude. However, it can be shown that 
equation (5) is still applicable if the component v z 
of the fluid velocity in the x direction is used in place 
of Ac. This seems a reasonable substitution, since it is 
only the component of the fluid velocity along the 
direction of the incident sound wave that affects the 
propagation of this wave. 

To compute k, then, integrals of the form 

> x dydz (12) 

must be evaluated. If the integrals over y and z are 
computed first, it is easy to see that the entire inte- 


J sin (47 rx\)dx j'J i 











SCATTERING 


483 


gral vanishes. The integral of v x over the yz plane is 
simply the net rate at which the fluid is flowing across 
this plane. At any time, the total amount of fluid 
passing through the yz plane in one direction must be 
just equal to the amount of fluid passing through in 
the other direction, and the net flow vanishes. Thus, 
a random distribution of velocity does not contribute 
to backward scattering of sound. However, sound 
may be scattered in other directions, as indicated in 
reference 3. 

Measurements at San Diego 4 and at Orlando 5 ' 6 
are consistent with the result that the sound scat¬ 


tered backward from velocity microstructure is very 
weak. At San Diego attempts were made to obtain 
echoes from underwater vortex rings, while at Or¬ 
lando a mechanical device was used to produce 
turbulent water in the path of a sound beam and at¬ 
tempts were made to measure the reflected sound. In 
both cases, no reflected sound could be observed. Al¬ 
though the data do not exclude the possibility that 
weak echoes may have been present , the combination 
of measurements and theory point to the conclusion 
that backward scattering of sound from velocity 
microstructure may be practically neglected. 



Chapter 30 


TECHNIQUE OF WAKE MEASUREMENTS 


M ost of the measurements of submarine and 
surface vessel wakes discussed in Chapters 2(5 
to 35 have been made by University of California 
Division of War Research [UC'DWR] or by Navy 
observers at the U. S. Navy Radio and Sound 
Laboratory [USNRSL] in San Diego. The instru¬ 
ments and physical principles applied to acoustic 
observations of wakes do not differ essentially from 
those employed in other underwater sound measure¬ 
ments described in Chapter 4, Chapter 13, and 
Chapter 21. It is unnecessary, therefore, to introduce 
here detailed descriptions of instruments and their 
theory. But before discussing the results, some general 
features of the experimental work at San Diego on the 
acoustic properties of wakes will be reviewed. 

30.1 LISTENING ANT) ECHO RANGING 

Listening through a wake to a ship under way, or 
to a mechanical noisemaker, constitutes the simplest 
type of acoustic observation of a wake. The presence 
of a wake manifests itself by a reduced sound level at 
the receiving hydrophone, compared with the same 
level with no wake interposed. Such observations of 
the acoustic screening effect are the incidental result 
of numerous measurements of the sound output of 
ships. But, in order to obtain quantitative results, 
it is desirable to use as sound source a transducer or 
mechanical noisemaker of constant power output, 
instead of the noise from the screws of a ship. To¬ 
gether with a hydrophone of constant sensitivity, 
this equipment makes possible determination of the 
transmission loss which sound undergoes in passing 
through a wake. 

Echoes returned by wakes can be studied by lis¬ 
tening or by using objective records of the current 
generated in the receiving channel of the transducer. 
While the second method is indispensable for the 
determination of sound intensities, it does not tell 
anything about the small changes in frequency that 
are caused by the relative motion of target and trans¬ 


ducer. The acoustic doppler effect is helpful in dis¬ 
tinguishing between the echo from a wake, which is 
nearly stationary, and the echo from the wake-laying 
vessel. This distinction is occasionally of practical 
interest, as in the study of the rather weak wakes 
produced by submarines in submerged level runs. In 
such cases it may be useful to preserve an audible 
record, in the form of a phonograph record, of the 
wake echo. The supersonic echo obtained aboard the 
experimental vessel is transmitted by short-wave 
radio to the laboratory ashore, where the phono¬ 
graphic recording can be done more conveniently 
than on a rolling and pitching vessel at sea. 

30.1.1 Sound Range Recorder Traces 

At San Diego it is a standard procedure in all wake 
work to secure echo records with a sound range 
recorder of the type in general tactical use. These 
chemical recorder traces are highly useful for a rapid 
estimate of the range of the wake and of the decay of 
its strength. As the chemically treated recording 
paper is unrolled, with the machine open, the ob¬ 
server makes pencil notes on the margin of the record 
concerning the work in progress, such as the begin¬ 
ning and ending of the oscillographic recording, 
changes of the sound frequency used, and other de 
tails. Thus the chemical recorder traces also provide 
a graphical log of the operations. 

The general appearance of wake echoes on the 
sound range recorder paper is illustrated by the 
photographic reproductions of original records shown 
in Figures I and 2. They are records of wakes laid 
by the auxiliary yacht E. W. Scripps between the 
echo-ranging vessel, the USS Jasper (PYcl3) and a 
target sphere buoyed at a center depth of (5 ft below 
the surface, in the course of experiments described 
in detail in Sections 31.2 and 32.3.2. The lower part of 
Figure 1 shows the sphere echo alone. Immediately 
after passage of the Scripps through the sound beam, 
there appears a strong wake echo and the strength 


484 


LISTENING AND ECHO RANGING 


485 


UJ 

2 

t- 




— WAKE | 
. ECHO 


l ©•<•>' 




7> 


. - 


V 


SCREW NOISE 
E.W.SCRIPPS passing; 


SPHERE ECHO 


RANGE 


Figure 1. Sound range recorder traces of wake echoes 
from E. W. Scripps. 


of the sphere echo is markedly diminished by the 
two-way transmission loss in the wake. Note the 


UJ 

2 


V 

-fed 


> 


PROJECTOR GRADUALLY 
TRAINED AWAY FROM 
WAKE, THEN TRAINED 
BACK. 


J 

\ 


PROJECTOR TRAINING 
CONSTANT 

•WAKE ECHO 

SPHERE ECHO 


J 


RANGE 


Figure 2. Sound range recorder traces of wake echoes 
from E. W. Scripps. 


gradual widening of the wake toward the top of the 
figure, as the wake grows older. 

The wakes were laid at right angles to the line 
connecting the transducer on the Jasper with the 
target sphere. In Figure 1 the projector was kept 
trained at the sphere in order to study the decay 
of a fixed part of the wake. Figure 2 shows the effect 
of gradually changing the training of the projector 
from its normal training; the range toward the near¬ 
est boundary of the wake increases, and since the 
sound beam now cuts obliquely through the wake, 
the apparent width of the wake increases propor¬ 
tionally to the secant of the angle included between 
the sound beam and the normal to the wake. On 





































TECHNIQUE OF H AKE MEASUREMENTS 


480 



Figure 3. Fathometer record of echoes from ocean surface. 


training back the transducer, the effect is reversed, 
thus causing a symmetrical pattern to appear in 
Figure 2. 

30.1.2 Fathometer Records 

Records of a wake, indicating its thickness and 
transverse structure, are readily obtained with a 
fathometer carried across the wake by a survey 
vessel. Such records may be utilized also to compute 
the vertical transmission loss, as long as a record from 
a standard target observable through the wake — 
for instance, the ocean bottom or surface — is also 
available (see Section 32.3.3). 

Early experiments were carried out with the fathom¬ 
eter mounted in the orthodox manner on a surface 
vessel. Records of the ocean bottom are then blanked 
out in certain cases when the survey boat enters a 
surface ship wake. This technique suffers from several 
disadvantages. It does not give an accurate value for 
the depth of the wake, since the duration of the wake 
echo is affected by the beam width, the pulse length, 


and other factors as well as by the depth of the wake. 
Also, the method is not very suitable for the measure¬ 
ment of the transmission loss through the wake, be¬ 
cause it requires the echo-ranging vessel to operate 
in relatively shallow water in order to record the 
bottom; furthermore, the depth and bottom char¬ 
acter may vary considerably while this vessel is 
moving. If, however, the fathometer is used in the 
inverted manner, by mounting it on the deck of a 
submerged submarine, those disadvantages are elimi¬ 
nated; clear strong records are obtained both of the 
highly reflecting ocean surface and of the surface 
ship’s wake, as illustrated by Figures 3, 4, 5, and 6. 

Figure 3 shows a record obtained while the sub¬ 
marine was diving from the surface. The depth scale 
marked 5 to 50 applies to this dive, with the time 
axis running from the right to the left. It can readily 
be verified from the double record in the center of the 
illustration that the weaker second reflection corre¬ 
sponds to depths that are exactly twice the depth of 
the stronger first reflection. Thus, the double record 





















































































LISTENING AND ECHO RANGING 


487 



Figure 4. Fathometer record of wake echoes from Coast Guard cutter Ewing. 


is a result of the sound traveling to the ocean surface 
twice and returning again to the submarine. The 
dark streaks at the top of this figure result from the 
acoustically reflecting region formed behind the 
submarine conning tower, presumably as a result of 
cavitation originating around the conning tower. 
The record at the far left is that of the ocean surface 
after the submarine arrived at a depth corresponding 
to the scale limit of the recorder and the scale was 
shifted to bring the record nearer the center of the 
paper. The small indentations and undulations of the 
record are produced by the surface swells. 

Reflection from a surface ship wake under which 
the submarine is passing produces in these records a 
shaded area protruding below the ocean surface, as 
shown in the next three illustrations. Figure 4 repre¬ 
sents the record of a wake laid by the USCGC Ewing r, 


proceeding at 13 knots. The submarine in this case 
passed under the wake at a point 350 yd behind the 
Ewing. This record was suitable for transmission loss 
calculations, according to the principles which will be 
described in Section 32.3.3. The result was a trans¬ 
mission loss of 42 db with 21-kc sound traversing the 
wake twice. Note that as a result of the large trans¬ 
mission loss, the record of the ocean surface is almost 
blotted out in the center of the wake, which had a 
thickness of 15 ft. 

The same effect is apparent in Figure 5, showing a 
wake record originating from the destroyer, USS 
Hopewell (DD681), proceeding at 10 knots. The 
distance astern is not accurately known, but it is 
roughly several hundred yards. The transmission 
loss at 21 kc through the center of this wake was 
32 db for the double path. The cause of the extrane- 





















































































488 


TECHNIQUE OF WAKE MEASUREMENTS 



• •• IVe « 


f=T • * » 


Figure 5. Fathometer record of wake echoes from USS Hopewell (DD681). 


ous markings on this record is uncertain; probably 
they are of instrumental origin. The wake is seen to 
be 30 ft thick at the maximum point. 

Figure 6 contains two records of the wake (17 and 
11 ft thick, respectively) of the Ewing, proceeding at 
13 knots; these records were not suitable for trans¬ 
mission loss calculations, since the amplification was 
increased to record the cross-sectional geometry of 
the wake. Comparison of Figures 4 and 6 gives an 
idea of the variations of wake structure occurring in 
practice; the vessel and speed are the same for both 
figures. For the proper interpretation of these cross 
sections, it should be remembered that the sound 
beam of the customary fathometer is rather broad, 
including an angle of about 30 degrees, thus causing 
the fine structure of the cross section to be smoothed 
out. 

30.1.3 Oscillograms 

In order to obtain permanent sound intensity 
records suitable for quantitative measurements, the 


current generated in the hydrophone is amplified and 
fed into a cathode-ray oscilloscope, the screen of 
which is photographed continually by a high-speed 
camera on standard moving picture film, as described 
in Section 4.3.3, Section 13.1.1, and Sections 21.2.1 
and 21.3.1. The developed negative shows a con¬ 
tinuous trace, representing the varying displacement 
of the luminous spot from its normal position on the 
oscilloscope screen. Time marks are photographed at 
suitable intervals as the film moves along steadily. 
By appropriate design of the electric circuits the dis¬ 
placement of the oscillographic trace is made propor¬ 
tional to the amplitude of the incident sound wave. 
The square of the amplitude of the oscillographic 
trace, therefore, is proportional to the intensity of the 
sound wave, at the face of the hydrophone, multi¬ 
plied by a factor depending upon the directivity of 
the hydrophone. If the sensitivity and the directivity 
pattern of the hydrophone are known, the scale of 
ordinates on the oscillogram can be calibrated in 
absolute units to yield the sound pressure in dynes 
per square centimeter. 






























































LISTENING AND ECHO RANGING 


489 



Figure 6. Fathometer record of wake echoes from Coast Guard cutter Ewing. 


This type of recording, which has been used widely 
in other sound studies, has usually been applied only 
to the analysis of wake echoes rather than to signals 
transmitted through wakes. The linear distance on 
the film from mid-signal to mid-echo provides a con¬ 
venient record of the range from which the echo was 
returned, since the distance on the horizontal scale is 
the product of sound velocity times the time. A 
number of oscillograms of wake echoes are repro¬ 
duced below on the scale of the originals. Figure 7 
shows three sets of three successive signals, each 3 
msec long, and the corresponding echoes both from a 
wake, laid by the E. W. Scripps, and from a target 
sphere 3 ft in diameter suspended behind the wake at 
a center depth of 6 ft. The oscillograms were obtained 
with 24-kc sound during Run 1 of the experiments 
summarized in Figures 8 and 9 of Chapter 31 and in 
Table 2 of Chapter 32, which should be consulted for 
a detailed description of the plan of observations. 

The numerical evaluation of wake oscillograms has 


so far been restricted to the visual measurement of 
peak amplitudes, described in Section 21.3.1, which 
generally have been held to be sufficiently repre¬ 
sentative of the echo as a whole. A more satisfactory 
though very time-consuming method would be to 
measure the amplitudes along the entire echo profile, 
square the amplitudes and integrate them over the 
time. This integral would be proportional to the total 
energy contained in the echo. It is possible to design a 
mechanism which would perform automatically this 
sequence of procedures. In any event, it would be 
desirable to supplement and check fundamental wake 
studies based upon measurement of peak amplitudes 
by investigating the total energy of echoes. 

Current procedure is to place the processed film on 
an illuminated viewer, read the peak amplitude of 
the echo with the aid of a transparent scale, and cor¬ 
rect the measured amplitude, if necessary, for the 
finite width of the luminous spot on the oscillograph 
screen. Averages over five successive echoes are 







































490 


TECHNIQUE OF WAKE MEASUREMENTS 



PR WS 

PR 

WS 

PR WS 





P ' 



^ ' r " n " 

P = PING 

W = WAKE ECHO 

R=REVERBERATION 

S = SPHERE ECHO 


Figure 7. Oscillograms of wake echoes from E. W. Scripps. 


taken, and the averaged peak amplitude is squared to 
obtain the echo intensity. The resulting average is 
different both from the average peak echo intensity 
and the average of the intensity over the entire echo. 
Since the spread of peak amplitudes may be as much 
as 10 db, this difference may be appreciable. The 
difference between average peak amplitudes and 
average intensities is discussed in Section 34.3.1. 

Finally, from the measured peak amplitudes the 
echo strength is computed according to the formula : 

E — S = 20 log A e — 20 log k , 

where E is the echo level in decibels above 1 dyne per 
sq cm, S the source level, defined as the sound level 
1 yd from the projector on its axis, also in decibels 
above 1 dyne per sqcm, and A e is the average peak 
amplitude of the echo as measured on the oscillo¬ 
gram. The constant k on the right side of this equa¬ 
tion has to be determined by calibration of the re¬ 
ceiving equipment; specifically, k is the amplitude 
measured on the oscilloscope with an incident wave 
whose pressure is 1 dyne per sq cm and with the same 
receiver gain at which .1,, is recorded. To determine 
S and k, an auxiliary transducer of known power out¬ 
put and of known sensitivity is used. 


30.2 OPERATIONS AND MEASUREMENTS 

Besides the acoustic measurements proper, ‘the 
study of wakes requires the determination of various 
auxiliary data. In the first place, the geometric co¬ 
ordinates of the part of the wake to which the 
acoustic data refer have to be known accurately. If 
the distance from the stern of the ship to the point 
where the sound beam strikes the wake is known, the 
age of the wake at the point of measurement may be 
found by dividing this distance astern by the speed 
of the wake-laying vessel and computations will be 
facilitated by use of Figure 3 in Chapter 35. Since 
the instrumental characteristics of the sound gear 
employed may undergo slow changes, it may become 
necessary to calibrate the gear immediately before or 
after the observation. Furthermore, there are a 
number of variable oceanographic factors whose in¬ 
stantaneous values have to be taken into account in 
interpreting the acoustic measurements. 

In addition to the wake-laying vessel, acoustic 
measurements on wakes require one vessel for echo 
ranging and an additional vessel when a transmission 
run is made in order to measure the horizontal trans¬ 
mission loss. For measurements of the transmission 























OPERATIONS AND MEASUREMENTS 


491 


loss, the use of a second experimental vessel carrying 
the receiver might be eliminated by echo ranging- 
through the wake at a target sphere and measuring 
the intensity of the echo returned to the transducer. 
From echo ranging at wakes, usually the wake-laying 
vessel proceeds at constant speed on a straight course 
past the measuring vessel, which either may run a 
parallel course with different speed or may be hove to. 
Maintenance of prescribed speeds and course de¬ 
mands accurate seamanship. The relative positions 
of the two vessels as a function of the time are de¬ 
termined by direct triangulation and dead reckoning. 
During echo-ranging experiments, an incidental check 
on those geometric data is obtained by the acoustic 
ranges. 

During transmission runs, the range from the cruis¬ 
ing auxiliary vessel, which carries the projector, to 
the measuring vessel has been accurately determined 
by the use of airborne sound; simultaneous radio and 
sound signals are transmitted from the auxiliary 
vessel, and the difference between the automatically 
recorded times of arrival of the two, multiplied by the 
velocity of sound in air, yields the range. Moreover, 
for transmission runs, the courses of the operating 
vessels have to be laid out and maintained with great 
care in order to avoid interference with the acoustic 
measurements from the auxiliary vessel’s own wake. 

In working with wakes which are laid and then 
allowed to age before the measurements or while the 
measurements are being conducted, the exact loca¬ 
tion of the wake soon becomes difficult to discern. 
If the wake-laying vessel lies to, it usually soon drifts 
enough to be useless as a marker for one end of the 
wake. The following method has proved helpful when 
working either with surface craft or submerged sub¬ 
marines, particularly with very long wakes. A small 
boat lies to at a point near where the initial end of the 
wake will be laid. As the wake-laying vessel goes by, 
the small boat moves into the center of the wake and 
releases a small amount of fluorescein ; a chrome yel¬ 
low or any other nonsoluble dye which floats on the 
surface is not satisfactory for this purpose, since 
wind drift can move it away from the wake. In the 
case of a submarine, it submerges as it passes the 
small boat or if already at periscope depth, the sub¬ 
marine releases the fluorescein. The wake-laying 
vessel releases fluorescein into the wake at the end of 
its run. Both this ship and the small boat then keep 

a Before using fluorescein in experiments at sea, it should be 
ascertained whether special authorization by the Area Com¬ 
mander is required. 


their bows touching the dye spot, thus keeping their 
net drift the same as that of the wake. More fluores¬ 
cein is dropped off the bow of the marker boats at 
intervals; one marking will not last when working 
with wakes older than 20 to 30 minutes. If the ob¬ 
serving vessel crosses the wake in the course of its 
measurements, the wake is located by sighting on the 
marker boats; the use of a simple optical device for 
lining up the markers is recommended. One of the 
marker boats takes a stadimeter range on the vessel 
which crosses the wake; this procedure aids in com¬ 
puting the wake age at that point. If successive cross¬ 
ings are made, fluorescein is dropped from the ob¬ 
serving ship just as it lines up the marker boats. The 
marker boat closest to this point then moves up to 
the new dye spot; this insures that the wake between 
markers remains free of extraneous wakes. Where 
marker boats are not available, it is helpful to use a 
mixture of fluorescein and chrome yellow as a marker. 
The two colors drift apart if any wind is present ; the 
chrome yellow can be seen farther away and is used 
to locate the fluorescein. 

30.2.1 Training Errors 

In echo ranging on wakes the trainable transducer 
is usually operated at a fixed relative bearing. How¬ 
ever, in measuring the transmission loss across a 
wake, it is necessary to keep the trainable projector 
of the sending vessel aimed at the hydrophone of the 
receiving vessel; continual changing of the bearing of 
the transducer is also necessary in echo ranging 
through the wake at a target sphere. For this purpose, 
an observer is stationed on the flying bridge to oper¬ 
ate a repeating pelorus, to be aimed at the auxiliary 
vessel or target sphere. A second man stationed at the 
control stack matches the projector-heading indi¬ 
cating “bug” to the target-bearing repeater. Even so, 
the projector heading does not hold precisely to the 
true target bearing. The deviation is partly attributa¬ 
ble to the lag in the various linkages of the system anti 
partly to the impossibility of holding the pelorus ac¬ 
curately on the target at all times. Under practical 
conditions as prevailing on board the USS Jasper 
(PYcl3), the estimated errors and their sources are as 
follows: 1 (1) pelorus aiming error +2 degrees in fair 
weather; (2) control stack matching error, projector 
bearing to target bearing + 2 degrees; (3) lag in train¬ 
ing system, gears and projector-heading repeater 
system ±2 degrees maximum prior to April 1944, 
when the system was overhauled and the error re- 




492 


TECHNIQUE OF WAKE MEASUREMENTS 


duced to approximately ± 1 degree. The maximum 
error is, therefore, ± 6 degrees and the probable error 
±3.5 degrees for data taken prior to April 1944, and 
±5 degrees and ±3 degrees, respectively, for data 
taken subsequently. 

At very short ranges there is another correction 
which may have to be applied because of parallax 
resulting from horizontal spacing between pelorus 
position and projector axis. On the Jasper this cor¬ 
rection amounts to 2.5 degrees for the aft projector 
and 3.5 degrees for the forward projector, when the 
target is 100 yd away and bears either 90 or 270 
degrees relative to the sending vessel. 

Finally there are training errors due to rolling and 
pitching of the sending vessel. This error can be 
serious at close range since rolls of 45 degrees have 
been experienced on the Jasper and rolls of over 20 
degrees are common in moderate weather. For the 
same vessel the pitching angle is of considerably 
smaller magnitude than that of the roll, rarely ex¬ 
ceeding 7 degrees. Installation of a device to record 
angle of roll, pitch, projector heading, target bearing 
and ship's heading for each sound pulse emitted, has 
been of great help in recognizing and rejecting 
acoustic observations that have been impaired bv 
serious training errors. 

30.2.2 Field Calibration 

The transmitting system’s absolute output has to 
be checked at the beginning and end of each day’s 
operation and also during the operation if excessive 
variations are encountered. For this purpose an aux¬ 
iliary transducer, whose performance is known from 
absolute calibration in the testing laboratory, is 
lowered into position by means of a special boom 
which pivots at the rail and swings down to projector 
depth. To check the actual output of the transducer 
in use, it is trained on the auxiliary transducer to 
give a maximum generated voltage; and from the 
laboratory calibration of the auxiliary transducer, the 
sound field pressure is determined. For the inverse 
calibration process, the known power output of the 
auxiliary transducer is received by the working trans¬ 
ducer and the generated current is recorded as in field 
work. Any appreciable deviation of any of the read¬ 
ings from those normally experienced requires an im¬ 
mediate investigation to determine the source of the 
difficulty. 

According to the experience of the San Diego group 
with the JK projectors, the standard deviation of the 


output pressure level was only 1.3 db over a period of 
15 months, and 0.4 to 1.0 db for groups of consecutive 
calibrations within that sequence. However, the 
standard deviation of the sensitivity of the receiving 
channel was 3.9 db for the same period and varied 
from 0.4 to 2.0 db for groups of calibrations within 
that period. The causes of this variation are un¬ 
known. In the course of one day, changes in overall 
sensitivity, which is the sum of the projector output 
and the sensitivity of the receiving channel, are 
negligible; changes in output level did not show any 
correlation with changes in receiver sensitivity. Also, 
over the whole period under discussion, changes in 
output level are not correlated — or at most are 
weakly correlated — with changes in receiver sensi¬ 
tivity. All these observations refer to 24-kc sound. 
Incomplete evidence suggests that the performance 
of 60-kc sound gear is even more variable. 

The existence of large calibration errors is also 
suggested by certain discrepancies among the San 
Diego data on the target strength of spheres, which 
are discussed in Section 21.4.3. 


30.2.3 Oceanographic Factors 

The weather and state of the sea appears to have 
some influence on the formation and gradual dis¬ 
solution of wakes. It is advisable, therefore, to keep 
a careful record of the circumstances prevailing at the 
time of the observations. The momentary oceano¬ 
graphic conditions have a profound effect also upon 
the propagation of underwater sound. Hence, it has 
become a standard practice to secure bathythermo- 
grams before and after each set of acoustic observa¬ 
tions. The transmission loss in the ocean intervening 
between sound gear and wake, which must be known 
in order to correct the measured data, is difficult to 
determine directly. So far, acoustic observations on 
wakes have not reached such a high degree of pre¬ 
cision as to make it imperative, as in the measure¬ 
ment of target strengths, to determine the transmis¬ 
sion loss in the ocean simultaneously with the wake 
observations. For details on the technique of meas¬ 
uring the transmission loss consult Chapter 4. 

Perhaps the most serious disturbances of under¬ 
water sound measurements are the rapid and un¬ 
predictable changes of the transmission loss, generally 
referred to as fluctuations and described in Chapter 7, 
which may amount to many decibels over intervals of 
only a few seconds. The only way to minimize their 



OPERATIONS AND MEASUREMENTS 


493 


influence is to take averages over long series of ob¬ 
servations. Even these averages may show a slow 
drift with time, sometimes called variation of the 
transmission loss, but the amplitude of the variations 
is of a lower order of magnitude than that of the 
fluctuations. In measuring the transmission loss 
which sound undergoes while passing across a wake, 
the fluctuations of the transmission loss in the sur¬ 
rounding ocean mask the effect sought after, or 
even may entirely obscure it for wakes in an ad¬ 
vanced stage of decay. In echo ranging, the sound 
returned by different parts of the wake undergoes 
destructive and constructive interference, which to¬ 
gether with the gradual change of the internal 
structure of the wake will invariably cause fluctua¬ 
tions of the wake echoes that are even more rapid 
than the fluctuations of the transmission loss. Conse¬ 
quently, wake echoes are even more variable in shape 
than sound signals which have been affected only by 


fluctuation of the transmission loss in the sea. Figure 
7 shows the irregular character of wake echoes re¬ 
sulting from changing interference effects. Fluctua¬ 
tions of the transmission loss in the ocean are also 
conspicuous in Figure 7; note the change in strength 
between the last two echoes from the target sphere, 
in the lower strip of the illustration. 

In echo ranging at wakes over short ranges, the 
reverberation background caused by the scattering 
of sound in the ocean constitutes an important limit¬ 
ing factor. Under conditions giving very high rever¬ 
beration, a weak echo may become lost in the back¬ 
ground. Strictly speaking, the echo intensities de¬ 
rived from measured echo amplitudes, as described 
above, include the contribution from reverberation 
and should be corrected for this superimposed inten¬ 
sity. In practice, this correction may usually be 
neglected whenever the wake echo is sufficiently 
strong to be distinguished from the reverberation. 



Chapter 31 


WAKE GEOMETRY 


I n this chapter information of rather heterogene¬ 
ous origin, concerning the dimensions of wakes, is 
brought together. Some types of acoustic observa¬ 
tions are in themselves eminently valuable for deter¬ 
mining the geometric characteristics of wakes. How¬ 
ever, a good deal has to be known about the geometry 
of wakes in order to plan and execute their investiga¬ 
tion by acoustic methods. Such knowledge has been 
provided by visual and photographic observations. 
Brief reference also will be made to thermal wakes, 
although very little is known about them so far. It is 
undecided whether or not the visual, acoustic, and 
thermal manifestations of the same wake agree as to 
the volume of the sea from which they originate; this 
problem deserves further study. 

31.1 WAKE GEOMETRY FROM AERIAL 
PHOTOGRAPHS 

The serial views of destroyer wakes shown in 
Figures 2 to 6 of Chapter 20 were selected from a 
large series of photographs, made available by the 
Photographic Interpretation Center, U. S. Naval Air 
Station, Anacostia. They show wakes of the destroyer 
USS Moale (DD693) proceeding on a straight course 
at constant speeds, ranging from 16 to 34.5 knots. 
For each speed, photographs of three or more differ¬ 
ent runs were measured, so that the results represent 
a fair average. The following conclusions are drawn 
from measurements made on the original prints. 

Immediately behind the screws the wake diverges 
with an included angle of about 50 degrees. Indi¬ 
vidual angles measured on different photographs vary 
between 40 and 60 degrees, but no clear-cut depend¬ 
ence on speed is indicated; these variations may well 
be spurious. It may be mentioned in passing that the 
wake of a stationary propeller 1 showed an angle of 
divergence of about 20 degrees. At a certain distance 
astern, the wide divergence of the destroyer wake 
ceases rather abruptly, and thereafter the wake 
spreads with a total included angle of about 1 degree. 


This angle too appears to be independent of the speed 
with which the wake is laid. However, the distance 
astern at which the transition from the 50-degree 
divergence to the 1-degree divergence occurs in¬ 
creases very markedly with the speed of the de¬ 
stroyer. At 16 knots, it is about 65 ft, and at full 
speed about 280 ft; this variation of distance with 
speed is not linear, as far as present experience indi¬ 
cates. The numerical values are given in Section 35.1. 
Observations of several different types, which will be 
reported in the rest of this chapter, all seem to indi¬ 
cate that the wake spreads out with a large included 
angle immediately behind the wake-laying vessel, 
and that at distances astern greater than about 100 
yd the wake spreads out with a very small included 
angle, of the order of 1 degree. However, none of 
these other observations have the same high intrinsic 
accuracy as the measurements on aerial photographs. 
Therefore, the results of these measurements, as in¬ 
complete as they are, have been selected for inclusion 
in Section 35.1. The large initial divergence of a wake 
is quite conspicuously demonstrated in Figures 1 and 
7 of Chapter 26, showing the wakes of a submarine 
chaser and destroyer, respectively. 

Aerial photographs also furnish interesting infor¬ 
mation on the cross-sectional structures of wakes. For 
instance, Figures 2 to 5 of Chapter 26 reveal that at 
short distances astern and at speeds less than 25 
knots the destroyer wake has a dense core and edges 
that stand out conspicuously; with increasing dis¬ 
tance, this internal structure gradually fades out. At 
speeds above 30 knots, the destroyer wake appears 
to be so strongly turbulent that the core is largely 
obliterated. 

Tran verse structure of a different kind is illustrated 
bv the submarine wakes seen in Figures 1 to 6. The 
wake of a surfaced submarine shows bifurcation, or 
twin structure, both at 15 and 20 knots in Figures 3 
and 4. The same illustrations clearly differentiate a 
short wake section immediately behind the sub¬ 
marine, which has a large angle of divergence, from 


494 


HATE OF WIDENING 


495 



Figure 1. Wake of surfaced submarine at 6 knots. 

the long wake proper, whose edges show little diver¬ 
gence. Thus, the general wake contour is quite similar 
for destroyers and surfaced submarines. 

Figure 7 gives a remarkable aerial view of a PT 
boat and its wake. The wake proper is narrow and 
compact, without visible structure, but the bow 
wave, for a distance astern of several ship lengths, is 
visually much longer than the wake. 

31.2 RATE OF WIDENING 

The rate of widening of an acoustic wake can be 
determined by measuring the gradual increase of the 
duration of the echoes obtained with a horizontal 
sound beam, as long as the signal length is much 
shorter than the wake width. In practice, subtracting 
the signal length from the measured length of the 
echo will correct for the prolongation of the echo 
length due to the finite signal length and will make 
possible a direct determination of the wake width. 

An analysis along these lines was made for four 
wakes laid by the E. W. Scripps on November 28, 
1944. The Scripps passed between the echo-ranging 
vessel, the USS Jasper (PYcl3), which was hove to, 
and a target sphere 3 ft in radius buoyed at a center 


Figure 2. Wake of surfaced submarine at 10 knots. 

depth of G.5 ft. The wakes were laid at right angles to 
the line connecting the sphere with the Jasper, each 
run being made in a new location of undisturbed 
water. All echoes were recorded oscillographically 
and sound range recorder traces were obtained simul¬ 
taneously. The signals consisted of pulses of 0.5, 1, 
and 3 msec long, transmitted in cyclic succession. 
The duration of the 3-msec echoes was measured 
both on the oscillograms and on the recorder traces; 
the results, expressed in yards, are plotted in Figures 
8 and 9 as functions of the time elapsed since the 
Scripps passed. The slope of these curves is the rate 
of widening. In order to find the width at any time, 
the plotted values of the echo length should be 
diminished by 2.4 yd. 

The average rate of widening of the Scripps wake, 
up to the age of 10 minutes, is 5 yd per minute for the 
chemical recorder traces, and 6 yd per minute for the 
oscillograms. This difference of 1 yd per minute can 
hardly be regarded as significant, in view of the dif¬ 
ferences between the several runs. Note that in both 
illustrations the graph of Run 2 is located between 
5 and 10 yd above the graph of Run 1, though both 
runs were made with 24-kc sound. The origin of this 
shift remains obscure, as the sea was unusually calm, 






WAKE GEOMETRY 


496 



Figure 3. Wake of surfaced submarine at 15 knots. 


almost without ripples, during the entire day, and the 
Scripps maintained the same speed of 9.5 knots dur¬ 
ing all four runs. Evidently, the geometric and 
physical properties of wakes are difficult to reproduce 
in repeated experiments, even under ideal weather 
conditions. The wake laid in Run 2 gave distinct 
60-kc echoes at an age of 30 to 40 minutes; during 
this period the wake width measured on the sound 
range recorder trace increased from 82 to 100 yd, 
corresponding to a rate of widening of about 2 yd 
per minute. No similar tests for the persistence of 
wakes were made during the other runs. 

Comparison of Figures 8 and 9 reveals that there 
is no systematic difference between the widths found 
for different sound frequencies. In particular, the 
alternating use of 45 and 60 kc during Run 4 gave 
results which are mutually consistent and agree 
quite well with the 24-kc graphs. 

At the stated speed of the Scripps, approximately 
300 yd per minute, a rate of widening of 5 to 6 yd per 
minute means that the total angle of divergence of 
the wake is about 1 degree. This figure is in excellent 
agreement with the angle of divergence found for 
destroyer wakes from aerial photographs. If the 



Figure 4. Wake of surfaced submarine at 20 knots. 

Scripps wake had the same great initial angle of 
divergence (about 50 degrees) as the destroyer wakes, 
it could not have been discovered by acoustic width 
measurement, because this method lacks the neces¬ 
sary “resolving power” along the time axis. At wake 
ages greater than 10 minutes, the rate of widening 
appears to decline steadily, and the angle of diver¬ 
gence must decrease correspondingly. However, it 
should be remembered that these observations were 
made on a calm sea. 

The rate of widening of thermal wakes can be 
studied by carrying a sensitive thermocouple across 
the wake at increasing distances astern. These in¬ 
vestigations are still in an exploratory stage, but 
they are mentioned here because preliminary results 
have been reported for two wakes laid by the E. W. 
Scripps. 2 Thus a comparison of the thermal and 
acoustic dimensions of the wakes laid by the same 
vessel became possible. Between the ages of 10 and 
60 minutes, the thermal wakes of the Scripps showed 
a linear increase in width from 30 to 50 yd. The rate 
of widening is about 0.4 yd per minute and the speed 
of the Scripps was 6 knots, or 200 yd per minute. 
Hence, the angle of divergence of the thermal wake 









FATHOMETER STUDIES 


497 




Figure 5. Wake of submarine during crash dive. 


Figure 6. Swirl behind submarine after crash dive. 


is only about 0.1 degree, or one-tenth of the diver¬ 
gence of the acoustic wake. In addition, the thermal 
wake appears to be much narrower than the acoustic 
one. 

Since the thermal and acoustic measurements were 
not made on the same day, it is by no means certain 
that the thermal and acoustic wakes behave as 
differently as these observations would seem to sug¬ 
gest. The 1-degree divergence found acoustically ap¬ 
plied to wakes less than 10 minutes old, while the 
available thermal data were apparently all for wakes 
more than 10 minutes old. Furthermore, the rate of 
widening may possibly depend on oceanographic 
factors, such as the temperature gradients in the 
surface layers of the sea. 

31.3 FATHOMETER STUDIES 

At the U. S. Navy Radio and Sound Laboratory 
[USNRSL], numerous measurements with a record¬ 
ing fathometer have been carried out on the wakes of 
a number of different surface vessels and submarines. 3 
Some of these wakes were investigated systemati¬ 
ca lly, and the width and depth of the wake was de¬ 


termined as a function of the distance from the wake- 
laying vessel and of its speed. 

31.3.1 Surface Vessel Wakes 

With surface ships, two methods were used for 
measuring the wake width. When ranging on the 
wakes of ships which happened to be passing, the 
survey boat, carrying the fathometer, crossed the 
wake as nearly perpendicularly as could be judged 
while speed and distance measurements were made. 
Some inaccuracy arose in the judgment of the angle 
of crossing when very far behind the wake vessel. 
However, the error introduced into the measured 
width by assuming perpendicular crossing was usu¬ 
ally negligible. When the survey boat was still farther 
astern, a greater error was present in determining the 
onset and disappearance of the wake record. The 
duration of recording was measured on chart paper 
(see Figures 3 to 6 in Chapter 30) by a caliper and 
rule. From the speed of the chart paper and of the 
survey boat, the width may be calculated. 

In the other method, which was suitable at close 
range when working with an assigned vessel, the 






498 


WAKE GEOMETRY 



Figure 7. Wake of PT boat at 25 knots. 


survey boat was towed by the vessel laying the wake; 
the latter is called the wake vessel. Measurements can 
then be made on a wake whose lifetime is effectively 
constant, in other words, for a constant boat speed, 
the survey boat is in a wake of the same age at all 
times. While the wake vessel maintained a steady 
course, the survey boat under tow was moved in and 
out of the wake on either side by using the helm. At 
the moment the survey boat passed the wake edge, 
as indicated by the fathometer record, the record was 
marked and a signal was sent to two observers on the 
wake vessel. One of these observers was on the stern 
and followed the transverse movement of the survey 
boat with a pelorus. At the instant of signaling, the 
angle of the fathometer mounting on the survey boat 
relative to the axis of the wake ship was noted. The 
other observer was on the bridge, and at the signal he 
instantaneously observed the ship’s compass course, 
for use in correcting for the angle of yaw of the wake 
vessel. The record was marked at the time of sig¬ 
naling the observers on the wake vessel, so that the 
data coidd be discarded if it were found that the 
survey boat was not exactly at the wake edge. 

The wake of the Jasper (overall length 127 ft, 
draft 12 ft, beam 23 ft) was studied by the second 


method over the range from 50 to 500 ft astern. All 
the measured widths, expressed in feet, agree with the 
formula 

w — w 0 + 0.0625^, (1) 

where Wo is the extrapolated width at the stern of the 
wake vessel, v the speed of the ship in feet per second, 
and t the time in seconds since its passage. By dif¬ 
ferentiating this formula with respect to the time, 

1 dw „ „ „ 

-= 0.0G25 = sin a , (2) 

v dt 

where a, the total angle of divergence of the wake 
edges, is 3.5 degrees. For similarly small distances be¬ 
hind destroyers, the wake edges were found to include 
an angle of about 50 degrees (see Section 31.1). The 
conspicuous discrepancy between this value and the 
corresponding one for the Jasper is doubtless due to 
t he different type of construction of these ships. The 
extrapolated value w 0 = 10 ft, in formula (1), is very 
nearly one-half the ship’s beam. At distances greater 
than roughly 5 ship lengths, the divergence of the 
Jasper's wake ceased at a width of perhaps two and 
a half times the ship beam; only random measure¬ 
ments by the first method were available for this 
region, however, and thus a small divergence angle of 
about 1 degree cannot be ruled out as far as large 
distances behind the Jasper are concerned. 

The measurement of wake thickness was carried 
out by proceeding into a wake and either remaining 
in its center while measuring distances and speeds, 
or by crisscrossing in order to investigate the thick¬ 
ness at points across the wake. Crisscrossing was 
necessary in order to locate the wake when operating 
at distances when the wake was not visible. A given 
wake will frequently have different acoustic trans¬ 
parencies at different points along its width. In some 
cases the thickness is the same along the width, and 
the greater transparency at the edge is caused by its 
less effective scattering properties. In other cases the 
wake is thinner at the edges. Some wakes are quite 
flat at the bottom, others are rounded at top and 
bottom, or may have one side which sinks below 
the other at both top and bottom. 

A wake cross section asymmetrical in the vertical 
plane parallel to the beam of the vessel was frequently 
observed and is apparently correlated with wind di¬ 
rection. Such records were first noted when operating 
with one engine of a twin-screw vessel and were 
thought to be the result of this asymmetrical source. 
The wake of a sailing vessel was investigated next, 
and was found to have an even more pronounced 





FATHOMETER STUDIES 


499 


100 


80 


V) 

o 

K 

< 60 
z 

X 

(- 

o 

S 








A {* 



A 




RUN 
24 K 

2 10 

/ / 

\ :' s/ 

t : 

\ ; 

4! 

JN 4 

5 KC 





RUN 4 

60 KC^"" 

v* .< 

''' H 

V $ 

\ /w 

TV/ 

V-V^RUN 
24 K 

L 

C 




y 

/ 

/ 

/ s 

/ f 

• ^ 

. / RUN 

■ VX '*~60 K 

3 

C 








/ / 











40 


* 

< 


20 


8 10 12 
AGE OF WAKE IN MINUTES 


14 


16 


18 


20 


Figure 8. Increase of wake echo duration for E. IF. Scripps at 9.5 knots. Measurements of oscillograms. 


100 


80 


v> 

o 

<r 

> 60 


x 

H 

o 

5 







RUN 4 

60 Kf\ 







RUN 2 

24 KCX, 











// 

// 

.''S 

- 

^ RUN 

S 45 K 

vRUN i 

24 KC 

4 

C 




RUN 3 

60 KCX. 

/> 

’^’^RUN 4 
60 KC 







XRUN 4 
45 KC 










< 


40 


20 


3 10 12 

AGE OF WAKE IN MINUTES 


14 


16 


18 


Figure 9. Increase of wake echo duration for E. W. Scripps at 9.5 knots. Measurements on chemical recorder traces. 


asymmetry. The sailing vessel heeled over consider¬ 
ably during the runs and the varying area of the hull 
in contact with the water on either side was con¬ 
sidered a cause for asymmetry. The wake of a twin- 
screw vessel when both screws were turning and when 
the wind was appreciable was then recorded. Again 


asymmetrical results were found. The effect is inde¬ 
pendent of the direction from which the survey boat 
crosses the wake. A typical value for the slope of the 
bottom of an asymmetric wake, as found for a 125- 
ft vessel, is 18 degrees. When the wind shifts from 
port to starboard, the cross-section geometry of the 











































500 


WAKE GEOMETRY 


wake should change to a mirror image of its former 
geometry, but in the majority of cases this expecta¬ 
tion is not entirely confirmed. Perhaps some as yet 
undiscovered parameter is responsible for this puz¬ 
zling behavior. 


results from actual variations of the wake structure. 

The wake thicknesses were plotted as a function of 
the distance astern and examined for a possible 
systematic variation. The slope of the bottom of the 
wake up to 800 yd astern was found to be 5 minutes 


Table 1. Wake thicknesses. 



USS Rathburne 
(ex-DD113) 

USS Hopewell 
(DD681) 

USCGC Ewing 

Speed in knots 

10-12 

10 

13 

Thickness of wake in feet 
for average distance 
astern of 400 yd 

19.5 ± 3.4 

23.1 ± 3.6 

13.7 ± 3.3 

Range of thickness in feet 

12-26 

10-32 

8-20 

Ratio * of wake thickness 
to ship draft 

1.63 

1.85 

1.52 


* These ratios are smaller than those previously found 3 for incidental destroyer wakes, and 
are believed to be more accurate. 


Aside from the miscellaneous results just described, 
an attempt was made to investigate systematically 
the variation of the thickness h of the wakes of two 
yachts, the USS Jasper (PYcl3) and the E. W. 
Scripps, with distance astern up to 3,000 ft and with 
speed from 3.5 to 11 knots. For either vessel, no 
systematic changes of h could be noted. A fair average 
of all measurements of h was 1.70 times the draft, or 
2.9 times the screw depth for the Jasper , and 1.11 
times the draft, or 3.0 times the screw depth for the 
Scripps (overall length 104 ft, draft 12 ft, single 
screw 8 ft above the keel). 

Scattered measurements made on the wakes of 
numerous large surface vessels of all types gave an 
average ratio of thickness to draft of 2.02. The wakes 
of small craft appear to be relatively thicker, with a 
thickness to draft ratio of the order of four. The only 
wake depth shallower than the draft was from a 
carrier wake 4,000 yd from the ship. For a speed¬ 
boat, h appears to increase considerably with speed. 

All these observations were made with a fathometer 
ranging downward from a measuring boat in the wake 
being investigated. Later measurements 4 were made 
with a fathometer mounted on the deck of a sub¬ 
merged submarine, ranging upward at the surface of 
the ocean. This method, for several reasons men¬ 
tioned in Section 30.1.2, provided more accurate data 
than was possible with the former. The accuracy of 
the individual thickness determination is such that 
the range of wake thicknesses summarized in Table 1 


of arc (or 4 ft per 1,000 yd) upward for the USS 
Rathburne (ex-DDl 13) and 16 minutes of arc (or 14 ft 
per 1,000 yd) downward for the Ewing. In other 
words, the differential quotient of the thickness of the 
wake with respect to the time, which will be required 
in the later discussion of the decay rate of wake 
strength, has the following values as upper limits: 

1 dh . 

— — = —0.08 mm 1 for the Rathburne at 10 knots, 
h dt 


— — = 0.04 min 1 for the Ewing at 13 knots. 

h dt 

Additional information on the rate of widening of 
destroyer wakes is found in a report by UCDWR.-' 
Wakes were laid by three different modern destroyers, 
running past the E. IF. Scripps at 15 knots. The 
Scripps was hove to and recorded the sound level of a 
transducer carried repeatedly across the wake by a 
50-ft motor launch. The sound level records showed 
definite breaks whenever the source crossed what 
may be called the acoustic boundaries of the wake; 
the time between these breaks was multiplied by the 
speed of the launch to give the width of the wake, 
suitable allowance being made for the occasional 
crossing occurring as much as 30 degrees away from 
the perpendicular transit. The plot of the entire data 
collected in this manner (Figure 22 in reference 5) 
suggested to the experimenters that new wakes widen 
more rapidly than old ones, with a total included 





















FATHOMETER STUDIES 


501 


angle of 2 degrees observed as far as 500 yd behind a 
15-knot destroyer, and an included angle of 1 degree 
thereafter. However, a critical examination of the 
plot reveals such a large quartile deviation that the 
reality of the differentiation between new and old 
wakes seems somewhat doubtful. An included angle 
of 1% degrees for the entire range of observations, 
with the maximum distance astern of 2,500 yd, gives 
a fair representation of the plot. The extrapolated 
initial width of these destroyer wakes is roughly 
equal to the beam of the vessel, perhaps somewhat 
smaller. However, the observations do not cast any 
light on the very large initial divergence of destroyer 
wakes, revealed by aerial photographs, because the 
acoustic measurements did not extend to distances 
less than 100 yd astern. It should be noted that the 
angle of spread derived from the acoustic measure¬ 
ments (I 1 ,? degrees) is in fair agreement with that 
derived from aerial photographs (1 degree), as re¬ 
ported in Section 31.1. It is possible to attribute the 
difference of 1 3 degree between the two figures en¬ 
tirely to the inaccuracies inherent in the respective 
processes of measurement. 

31 . 3.2 Submarine W akes 

Information on the geometry of submarine wakes 
is less detailed. Among the measurements made with 
the fathometer ranging downward , 4 an investigation 


Table 2. Wake of submarine at periscope depth. 


Distance from 
periscope in yards 

Wake top 
in feet 

Depth of wake 
bottom in feet 

67 

39 

70 

100 

0 

27 

117 

0 

40 

152 

0 

36 

215 

0 

31 

315 

0 

25 

319 

0 

28 

350 

0 

30 

450 

0 

23 


of the wake of a fleet-type submarine 309 ft long is 
reported. At periscope depth the keel is submerged to 
a depth of GO ft, the screws to a depth of 48 ft, and 
the deck to a depth of 35 ft below the surface; the 
speed was 5.5. knots. Table 2 contains the observed 
depths. The same information for a surfaced sub¬ 
marine of the same class, moving at 7 knots, is given 
in Table 3. 


The maximum distance of 450 yd appearing in 
Table 2 is not the upper limit of detectability of the 
wake at a keel depth of 00 ft, as the observers 
emphasized. 

The length of the subsurface wake of an S-class 
submarine , 6 running at G knots, was found to be 
about 1,000 yd at a depth of 45 ft, 235 yd at a depth 
of 90 ft, and 100 yd at a depth of 125 ft. These 
figures give the distances astern of the submarine 
over which the wake extends before it becomes un¬ 
detectable by the gear used in these experiments. 
The bow-mounted 24-kc transducer was trained at 
a fixed bearing of 30 degrees relative to the Jasper, 


Table 3. Wake of surfaced submarine. 


Distance 
astern 
in yards 

Wake bottom 
depth 
in feet 


100 

32 


145 

24 


180 

29 


300 

26 


480 

18 


660 

26 


800 

21 


950 

22 



which was following the submarine on a parallel 
course and then gradually fell back. At creeping speed 
(2 to 3 knots) the length of the acoustically effective 
wake is less than 30 yd for a fleet-type submarine, 
according to recent San Diego observations . 7 It would 
seem, then, that the subsurface wake is not a good 
scatterer at greater than periscope depth, particularly 
at slow speeds. Analogous experiments 8 at frequen¬ 
cies of 24 and 45 kc were carried out with a fleet-type 
submarine, running at speeds up to 9 knots and at 
depths down to 400 ft. According to the observers, 
during no run was an echo definitely identified as 
coming from the wake alone. 

From the data summarized in Table 2, it appears 
that the wake of a fleet-type submarine, running at 
5.5 knots at periscope depth, extends to the surface at 
distances astern greater than 100 yd; the single record 
at a shorter distance of 67 yd, which suggested a com¬ 
pletely submerged wake, unfortunately was uncer¬ 
tain. Later tests, using the same fathometer equip¬ 
ment with the fleet-type submarine USS Trepang 
(SS412), have led to a general confirmation of the 
previous results . 9 The Trepang was running at 8 
knots at a keel depth of 60 ft, and the wake appeared 













WAKE GEOMETRY 


502 


at the surface at a distance of 600 ft astern. From 
this figure, the slope of the top of the wake may be 
computed, assuming it is constant; the ratio of screw 
depth to distance of emergence of wake, 48/600 or 
0.08, corresponds to a total angle of divergence of 
9 degrees at the screws. This value, however, is 


based on only one record. No clean-cut wake records 
were obtained at greater depths (200 and 400 ft), but 
this may be attributed to purely operational diffi¬ 
culties since the submarine found it difficult to pass 
directly under the stationary launch carrying the 
fathometer. 



Chapter 32 


OBSERVED TRANSMISSION THROUGH WAKES 


I n crossing a wake, sound undergoes a transmis¬ 
sion loss in addition to that resulting from propa¬ 
gation through the ocean at large. Transmission loss 
in the ocean is primarily geometric — the sound 
beam spreads over large distances because of the 
inverse square law and because of refraction condi¬ 
tions. At frequencies less than 100 kc, the transmis¬ 
sion loss from physical causes, such as scattering and 
absorption, is not very important at the short ranges 
— a few hundred yards or so — of interest in wake 
measurements. 

The observed transmission loss in wakes, however, 
is ascribed exclusively to physical causes, scattering 
and absorption by air bubbles, because the dimen¬ 
sions of wakes are much smaller than the distances 
over which the geometric effects are particularly im¬ 
portant. An exception to this rather sweeping state¬ 
ment may have to be made in the case of sound 
originating in the wake, as described in Section 32.3. 
These phenomena, however, are little understood at 
present, as they have not been sufficiently studied. 


32.1 DEFINITIONS 


The physics of the transmission of sound through 
wakes has already been fully discussed in Chapter 28. 
All that is necessary here is to summarize the conven¬ 
tions concerning the expression and presentation of 
the measurements of the transmission loss through 
wakes. 

The total transmission loss undergone by a sound 
beam on traversing a wake, or the attenuation, as it is 
usually called in underwater sound work, is defined 
by the equation 


77,„ = 


10 log 


/( 0 ) 
I(w ) ' 


(1) 


where 7(0) is the intensity of a parallel beam of 
underwater sound before entering the wake, and 
I(w) is its intensity after it has penetrated the entire 
width w of the wake; the transmission loss in the 


wake //„ is distinguished by the subscript w from the 
transmission loss in the ocean at large, which is com¬ 
monly denoted by the symbol 77. According to equa¬ 
tion (53) of Chapter 28, the attenuation by the wake 
can be represented as a product, namely 

H,„ = K e w , (2) 


where w is the geometric width of the wake, usually 
measured in yards, and K, is the so-called coefficient 
of attenuation in decibels per yard. Definitions (1) 
and (2), as they stand, apply to a sound beam im¬ 
pinging perpendicularly upon the wake; for oblique 
incidence w obviously has to be replaced by w sec 0, 
where /3 is the angle included between the beam and a 
line perpendicular to the wake. 

Note that equation (1) may be written in the form 


I(w) 

7(0) 


10 


H w /10 


or, by substitution from equation (2), 


I(w) 

7(0) 


ur*' 


10 


(3) 

( 4 ) 


Both 77and K e are overall properties of the wake, 
and it remains to express them as functions of the 
physical parameters describing the microstructure of 
the wake, which is known to consist of multitudes of 
bubbles of all sizes. The acoustic properties of bubbles 
have been characterized in Chapter 28 by their indi¬ 
vidual cross sections <r s , <r„, ov for scattering, absorp¬ 
tion, and extinction of sound, respectively. It will be 
remembered that these quantities vary considerably 
according to the size of the bubbles, and that, by and 
large, only bubbles near resonant size make a signif¬ 
icant contribution to the average cross section ap¬ 
plying to a population of bubbles of all sizes. 

Should all the bubbles in the wake happen to have 
exactly the same size, the coefficient of attenuation 
would be given by [see equation (55) of Chapter 28] 

K e = — = 4.34n<r e db per cm (5) 

w 


503 


504 


OBSERVED TRANSMISSION THROUGH WAKES 


or I\ e = — = 390.8 m e db per yd , (6) 

w 

where <r f in square centimeters is the extinction cross 
section of this particular size of bubble and n in cm -3 
is the average number of bubbles of this size per 
cubic centimeter in the wake, as defined by equa¬ 
tion (54) of Chapter 28. In the more realistic case of 
bubbles of many sizes, the attenuation coefficient is 
given by equation (67) of Chapter 28, 

K e = — = 1.4 X 10 b u(R r ) db per yd , (7) 

w 

where u(R)dR is the total volume of air contributed 
by bubbles with radii between R and R + <IR in 1 cu 
cm of the air-water mixture, or rather the average of 
this quantity taken over the entire column in which 
the sound beam and the wake intersect; R r in equa¬ 
tion (7) is the radius of the resonant bubbles cor¬ 
responding to the sound frequency used in deter¬ 
mining H w . 

The total attenuation corresponding to equation 
(5) is 

H w = 4.34<r e nw = 4.34cr e A T (u>) , (8) 

where N(w) — nw denotes the total number of 
bubbles in a column of unit cross section. Thus N(w) 
is a measure of the total bubble population affecting 
the sound beam. 

Differentiating equation (8) logarithmically with 
respect to the time, the decay of the transmission loss 
across the wake is obtained. 

clIU _ _L_ dN{w) 

H w dt ~ N(w) dt 

At first, dN(w)/dt perhaps will be positive for suf¬ 
ficiently small bubbles, whose number might be in¬ 
creased rapidly by the gradual dissolution of bubbles 
of originally larger size. But ultimately, dN(w)/dl 
must become negative. It is seen then that the decay 
rate of the transmission loss affords a direct measure 
of the rate of disintegration of the bubble population. 

32.2 EXPERIMENTAL PROCEDURES 

In principle, the experimental determination of the 
transmission loss through the wake requires only 
relative measurements of sound intensities. If over 
the period of observations the range from transducer 
to hydrophone, and the transducer output and hydro¬ 
phone sensitivity remain constant, then the absolute 
values of any of these three quantities does not have 


to be known; the difference of sound levels recorded 
by the hydrophone before and after the wake has 
been laid across the sound beam is simply the trans¬ 
mission loss H w . Whenever, during the course of ex¬ 
periments, the range changes appreciably a correc¬ 
tion must be applied, based on the appropriate value 
of the transmission loss H in the surrounding ocean. 
Care should be taken to place both transducer and 
hydrophone at such a depth that they are completely 
hidden from each other by the wake. 

A characteristic feature of transmission measure¬ 
ments of this simplest type is that, for sufficiently 
short wavelengths, only a very narrow cone of the 
divergent sound beam emitted from the transducer is 
utilized, namely the solid angle subtended by the 
face of the hydrophone at the location of the trans¬ 
ducer. Thus, the instantaneously recorded transmis¬ 
sion loss is for a sharply bounded layer of the wake. 
The roll and pitch of the vessel carrying the trans¬ 
ducer and hydrophone will raise and lower both 
instruments and will cause that narrow pencil of 
sound to traverse the wake at different depths below 
the ocean surface. Since the acoustic thickness of the 
wake is likely to vary somewhat vertically, corre¬ 
sponding variations of the measured transmission 
loss must be expected. 

In one respect these variations are even helpful. 
They afford an automatic smoothing out of the verti¬ 
cal variations of the acoustic thickness and thus pro¬ 
duce a better representation of the average state of 
the wake. The directivity of the sound gear is also 
important, in so far as rolling and pitching of the 
vessels carrying the transducer and hydrophone, to¬ 
gether with possible training errors, may affect their 
relative orientation and hence may cause fluctuations 
in the strength of the signals received. In practice, 
this effect cannot be separated from other fluctua¬ 
tions of the signals, resulting from changes of the 
transmission loss in the ocean interposed between 
transducer and hydrophone. By averaging over long 
series of signals, these disturbing influences may be 
minimized, though perhaps not fully eliminated. 

32.3 TRANSMISSION LOSS ACROSS 
WAKES 

32.3.1 One-Way Horizontal Trans¬ 
mission Loss 

Transmission loss in wakes has been investigated 
comprehensively only for five vessels of the destroyer 





TRANSMISSION LOSS ACROSS WAKES 


505 



<s 




0 50 100 150 200 250 300 

AGE OF WAKE IN SECONDS 

Figure 1 . Sound transmission loss due to wake versus 
age of wake. Ship IV, December 30, 1943, 15 knots. 
Source beyond wake. 


type — two old destroyers of the 1916-1917 class, a 
new destroyer of the Fletcher class, and two destroyer 
escorts. 1 Wakes were laid at speeds of 10, 15, 20, and 
25 knots. A 50-ft motor launch repeatedly carried the 
projectors, mounted at depths of 6 and 7 ft, respec¬ 
tively, across the wake, while the hydrophones were 
suspended from the bow of the E. W. Scripps at a 
depth of 10 ft, about half the depth of the wake (see 
Section 31.3). Sound at frequencies of 3, 8, 20, and 
40 kc was recorded both with the launch beyond the 
wake and with the launch inside the wake; while 
sound recorded when the launch was on the near side 
of the wake provided reference values. 

By applying a correction for the measured average 
transmission loss in the ocean, all sound levels were 
reduced to a standard distance of 100 yd, for the 
three cases of (1) source beyond wake, (2) source in 
wake, anti (3) no wake intervening. The difference 
between case (3) no wake intervening and case (1) 
source beyond wake was taken to be the transmission 


SOURCE IN WAKE 
EQUATION (10) 



SOURCE BEYONO WAKE 



SPEED OF WAKE-LAYING VESSEL IN KNOTS 


NOV. 

DEC. 

FREOUENCY 
IN KC 

A 

▲ 

3 

□ 

■ 

e 

O 

• 

20 

X 


40 


Figure 2. Dependence of transmission loss on speed of 
wake-laying vessel. 


loss for the source beyond the wake, or H w as defined 
in equation (3) of Section 32.1; similarly, the differ¬ 
ence between case (3) no wake intervening and case 
(2) source in wake was taken to be the transmission 
loss for the source in the wake. 

In the original paper, the results are reproduced in 
separate graphs for each of the several vessels, speeds, 
frequencies, and locations of the source; one of these 
is reproduced in Figure 1. However, not all the possi¬ 
ble combinations of the different parameters are 
actually shown. Although not representing the best 
fit for every single set of observations, the following 
interpolation formulas are believed to represent ade¬ 
quately most of the data. 

Source in wake H' w — 2A{vf) } - — (4.8 ± 1.6)£, (10) 

Source beyond wake H w = 1.5 (vfy — (3.0 ± 1.4)f, (11) 

where v is the ship’s speed in knots, / is the frequency 
of the sound in kilocycles and t is the time in minutes 
which has elapsed since the passage of the screws or 
age of the wake. No standard errors are assigned in 








































































OBSERVED TRANSMISSION THROUGH WARES 


506 


the original report to the numerical coefficients of the 
first terms of equations (10) and (11), but it is stated 
that the initial values (t — 0) of the transmission loss 
for individual runs show a scatter of the order of 3 db. 
Figure 2 gives an idea of the accuracy with which 
equations (10) and (11) represent the initial trans¬ 
mission loss at different speeds and frequencies. 

A higher transmission loss for case (2) source in 
wake, than for case (1) source beyond wake, appears 
to be well established observationally, but the theo¬ 
retical explanation for this systematic difference is not 
at all evident. With the source located inside the aer¬ 
ated water of the wake, air bubbles are likely to be 
held on the face of the transducer by adsorption. 
There are theoretical reasons for believing that such 
a layer of adsorbed gas should reduce, or “quench,” 
the output of the transducer, causing an apparent 
increase of the transmission loss in case (2). However, 
it is somewhat surprising that the quenching effect 
should show a behavior regular enough to follow 
equation (10). 

The difference in the decay rate for case (2) the 
source in wake and case (1) the source beyond wake 

is 

dH\, dH , o ,, 

■-— —— = 1.8 db per minute. 

dt dt 

This difference ma} r not be significant in view of the 
standard errors of these quantities. However, if it is 
accepted at its face value, the relative rates of decay 
are equal to each other, and given for fresh wakes by 
the equation 

1 dH w ^ 1 dH' w _ 2 
H w dt //'. dt ( vfy' 

This equality between the two rates is evident from 
equations (10) and (11) in which corresponding co¬ 
efficients have the same ratio of 5/8. According to 
equation (9) of Section 32.1, the relative rate of decay 
is a function solely of the rate of disintegration of the 
bubble population. The physical significance of the 
observed decay rate will be discussed in Section 34.4. 

Some incidental information on the transmission 
loss across wakes has been obtained during measure¬ 
ments of the underwater sound output at 5 kc of a 
destroyer, cruiser, and aircraft carrier 2 observed at 
varying speeds. Measurements at higher frequencies 
were also made, but the results are inconclusive as 
far as the transmission loss across wakes is concerned. 
All that can be said about the transmission loss at 25 
and 60 kc is that it is distinctly higher than at 5 kc; 


residual sound intensities, after passage through the 
wake, in most cases had dropped to the background 
noise and thus made impossible an evaluation of the 
transmission loss. Even the 5-kc data, plotted as a 
function of age of the wake, are rather widely scat¬ 
tered. But for each of these vessels the plot is not in¬ 
consistent with tentative predictions made from 
equation (11) above, a fact that is somewhat sur¬ 
prising in view of the dimensions of two of these three 
ships listed in Table 1. The initial transmission loss 


Table 1. Ship dimensions. 



Destroyer 
USS Colhoun 
(DD801) 

Light 
Cruiser 
USS Trtnton 
(CL11) 

Carrier 

USS Hancock 
(CV19) 

Length in feet 

376 

555 

874 

Beam in feet 

39 

55 

93 

Draft in feet 

13 

13 

29 

Screw depth in 
feet 

11.25 

19.5 

21.3 


(t = 0) is about 15 db at 5 kc for each vessel, while 
formula (11) gives 17 db at 25 knots for 5-kc sound. 
At higher frequencies, the greater absolute value of 
the transmission loss might facilitate the detection 
of possible differences between the destroyer and the 
larger ships. 

32.3.2 Two-Way Horizontal Trans¬ 
mission Loss 

Another method of measuring the horizontal trans¬ 
mission loss has been tried out in experiments aimed 
at a simultaneous determination of wake echo 
strength and transmission loss. 3 The E. W. Scripps, 
running at 9.5 knots on a straight course, laid a wake 
between the USS Jasper (PYcl3) and a target sphere 
3 ft in diameter buoyed at a center depth of G.5 ft. 
The drop in the apparent target strength after the 
wake was introduced thus was taken to represent the 
two-way transmission loss across the wake. The wake 
echo intensity could also be measured on each oscillo¬ 
gram, giving the effectiveness of the wake as a 
scatterer of sound. A plot of the sphere and wake 
echo levels for one of the 24-kc runs is shown in 
Figure 7 of Chapter 33. The results are summarized 
in Table 2; further reference to the decay rate of the 
echo strength will be made in Section 33.4. 

The 45 and 60-kc data on which the echo strength 

















TRANSMISSION LOSS ACROSS WAKES 


507 


recovery and decay rates quoted are based were 
taken quasi-simultaneously by tuning the sonar 
equipment alternately to the two frequencies for two- 
minute intervals. The wake and sphere distances for 
this run are those quoted in the 45-kc row. The 9-db 
drop in apparent target strength at GO kc is based on 
a separate run, with wake and target distances as 
stated in the third row of the table. 


32.3.3 Two-Way Vertical Trans¬ 
mission Loss 

A recording fathometer has been used for the meas¬ 
urement of sound transmission loss in the vertical 
direction through surface ship wakes. 5 The fathom¬ 
eter was secured on the deck of the submarine USS 
S-18 (SS123) so as to range upward onto the surface 


Table 2. Effect of wake on sphere echoes. 


Frequency 
in kc 

Distance to 
wake center 
in yards 

Distance 
to sphere 
in yards 

Depth at 
which sound 
beam passed 
through wake 
in feet 

Maximum drop 
in sphere 
echo level 
with wake 
present 
in db 

Rate of 
recovery 
of sphere 
echoes 
in db per 
minute 

Rate of 
decay of 
wake echoes 
in db per 
minute 

24 

270 

350 

10 

6 

1.4 

1.5 

45 

97 

162 

12 

No data with 

fresh wake 

0.7 

60 

58 

98 

12 

9 

0.8 

0.7 


Earlier measurements of the Scripps wake 4 gave 
a depth of the wake bottom of 13 ft. According to the 
values quoted in Table 1, the sound beam passed 
definitely above this bottom depth of 13 ft. However, 
a short time before the data of Table 2 were obtained, 
the Scripps had been outfitted with a new engine and 
propeller, so that the wake dimensions may have been 
altered to some extent. The present propeller is 3.8 ft 
in diameter and the shaft is 5.5 ft below water line. 
Therefore, since the Jasper’s sound projector is 15 ft 
deep, maximum acoustic shadowing of the sphere by 
the wake could not be expected immediately after 
the Scripps had passed. These wakes widened later¬ 
ally, as measured by the wake echo elongation, at 
about 6 yd per minute. The same rate of spreading 
may also be applicable in the vertical sense without 
necessarily implying that a strongly absorbent “core" 
of the wake ever moves down to an effective screening 
position in these experiments. This may account for 
the low magnitude of the observed transmission loss. 

Similarly the decay rate of the transmission loss 
dHu /dt is one-half the rate of recovery of the sphere 
echoes; hence dH w /dt is 0.7 and 0.4 db per minute for 
24- and GO-kc sound, respectively. These decay rates 
are much smaller than that of destroyer wakes, which 
were found to be independent of frequency — 3.0 db 
per minute. However, the relative rates of decay are 
in moderate agreement with those computed from 
equation (10) for a destroyer speed of 10 knots; 
numerical values are shown in Table 3. 


of the ocean, the echoes being continuously recorded 
in the control room. With this arrangement, the 
effect of a surface ship wake is recorded as the sub¬ 
marine passes beneath it. The ocean surface is used 
as a “standard target.” Sample records obtained 
with this method are shown in Figures 4 to 6 of 
Chapter 30. 


Table 3. Relative rates of decay. 



24 kc 

60 kc 

1 dH w „ 

77 —t— for E. IF. Scripps at 9.5 



knots (from Table 2) 

0.23 min- 1 

0.09 min- 1 

1 (IH 

- - for DD at 10 knots [from 



equation (10)] 

0.13 min -1 

0.08 min -1 


Quantitative transmission loss results are obtained 
from the fathometer records by a special procedure of 
operating the instrument in conjunction with calibra¬ 
tion records made in the laboratory. As the submarine 
passes beneath a surface ship wake, an attenuator in 
the receiver-amplifier is adjusted so that the effect of 
the wake plus the effect of the attenuator is such that 
a light gray “voltage-sensitive” record of the ocean 
surface echo is produced on the chart paper. Some 
practice is required, as very little trial-and-error time 
is available while the submarine is directly below the 
wake. 
























508 


OBSERVED TRANSMISSION THROUGH WAKES 


The procedure is completed by determining, in 
effect, the amount of amplifier attenuation required 
to record the unobscured ocean surface at the same 
density as that of the record taken below the wake. 
The difference of attenuator settings in the two cases 
yields the wake transmission loss directly. The actual 
procedure was complicated by the lack of a calibrated 
attenuator; the details of the necessary laboratory 
calibration of the gain control by matching records 
for different gain settings and echo levels are de¬ 
scribed in reference 5. The fathometer record yields 
an accurate value of wake thickness in each case so 
that attenuation coefficients can be computed in 
decibels per foot of wake thickness. 

The coefficient of reflection at the ocean surface 
cancels from the measured transmission loss, because 
n affects the sound levels both in and outside the 
wake in an identical manner, aside from slow varia¬ 
tions of the state of the sea. If the ocean surface were 
a perfect plane, and if the axis of the sound beam 
impinged upon it perpendicularly, the entire off-axis 
output of the fathometer would be reflected so as not 
to return to the transducer. On account of the waves, 
swells, and other irregularities of the surface, and be¬ 
cause of imperfect leveling of the submarine, actually 
some off-axis sound is reflected back on to the face 
of the transducer. Hence, it is necessary to keep the 
submarine at a depth shallow enough to make the 
central lobe of the sound beam fall entirely inside the 
wake. This condition was well fulfilled during these 
experiments, the results of which will now be 
described. 

Sound of 21 kc was found to undergo an average 
attenuation of 18 + 3 db during vertical one-way 
passage through the wakes about 400 yd behind the 
USS Rathburne (APD25, ex-DDll3), USS Hopewell 
(DD081) and USCGC Ewing, traveling at speeds of 
10 to 13 knots. Combining these total attenuations 
with the wake depths h for the vessels, accurately 
determined from the same records and already dis¬ 
cussed in Section 31.3, average attenuation coef¬ 
ficients in the vertical direction in decibels per yard 
could be computed and were found to be 3.0 ± 0.6 db 
per yd for the Rathburne and Hopewell and 4.8 + 1.5 
db per yd for the Ewing. These are grand averages, 
disregarding differences in the distance astern, which 
are unknown in many cases, and disregarding devia¬ 
tions of the point of measurements from the center of 
the wake; moreover, some “knuckles” are included 
with the straight runs. If only data referring to 
known distances astern and to the center of straight 


wakes are retained, all observations applying to 
wakes laid by the Hopewell are eliminated. Plotting 
as a function of the distance astern, the attenuation 
coefficients for the wake of the Rathburne, running at 
a speed of 10 knots (corresponding to a screw-tip 
speed of 52 ft per sec), the following linear interpola¬ 
tion formula is found for the range from 100 to 
800 yd: 


Hw 

h 


(3.135 + 0.057) -(- (0.093 ± 0.018) X 


distance astern 
100 yd 


db per yd . (13) 


Since there is apparently no correlation between the 
total transmission loss in the wake H w and the dis¬ 
tance astern, equation (13) implies that the wake be¬ 
comes thinner in the vertical direction as it ages. A 
similar plot for the Ewing, running at 13 knots, re¬ 
veals an enormous variation of the attenuation 
coefficient ranging from 2.4 to 6.6 db per yd without 
any clear dependence on the distance astern. The 
distances astern cannot, however, be very accurately 
determined in these experiments. There is no obvious 
explanation why the Ewing data should show a 
greater scatter, enough to obliterate any dependence on 
distance astern. The higher value of the attenuation 
coefficients for the Ewing has been associated tenta¬ 
tively with the greater screw-tip speed (112 ft per 
sec at 13 knots) of this vessel, compared with the 
two destroyers. No corroboration for this surmise 
could be found among the observations of the hori¬ 
zontal transmission loss through wakes laid by dif¬ 
ferent destroyers, already described in Section 32.3.1, 
although the screw-tip speeds of these vessels ranged 
from 80 to 137 ft per sec at 15 knots, and from 53 to 
95 ft per sec at 10 knots. 

It is of interest to compare the attenuation coef¬ 
ficient in the vertical direction with that computed 
from the total transmission loss measured hori¬ 
zontally. According to equation (11) in Chapter 32, 
the one-way horizontal transmission loss through the 
wake 400 yd behind a destroyer traveling at 10 knots 
is about 20 db for 21-kc sound. The width of this 
wake is about 75 ft, according to Figure 22 of refer¬ 
ence 1. Hence, the horizontal attenuation coefficient 
is about 0.9 db per yd, or about one-third of the 
vertical one, as reported above for the destroyers 
Rathburne and Hopewell. This discrepancy is probably 
real, but hardly disturbing. In fact, the average at¬ 
tenuation coefficient would be expected to be smaller 




PROPAGATION ALONG WAKES 


509 


horizontally than vertically in case the wake has a 
strong core and weaker fringes, because the vertical 
measurements refer to the center of the wakes. 

32.4 PROPAGATION ALONG WAKES 

On the whole, the methods employed for the study 
ot sound propagation across wakes, described in 
Section 32.3, have led to apparently consistent re¬ 
sults. For sound propagation along wakes, however, 
the observations do not fit easily into the general 
picture; they are a few in number and provide in¬ 
sufficient data to permit a complete analysis of all the 
factors involved. 

A mechanical noisemaker s was towed both in and 
below the wake of a destroyer running at 10 and 14 
knots, and sound levels were recorded simultaneously 
by two hydrophones — one towed in the wake by the 
destroyer and the other suspended at a depth of 10 ft 
from a boat which was hove to. The destroyer fol¬ 
lowed a straight course past this boat, while the 
distance between the noisemaker and the towed 
hydrophone was steadily increased from 50 ft to 
1,200 ft by unreeling the hydrophone cable. 

As the cable lengthened the hydrophone gradually 
descended, ultimately passing below the bottom of 
the wake, which was assumed to be 20 ft below the 
surface. 5 The noisemaker was towed 50 ft behind the 
destroyer, and the hydrophone reached a depth of 
20 ft at distances of 400 ft (10 knots) and 1,000 ft (14 
knots) behind the noisemaker. 

Finding the transmission loss along the wake would 
require comparing the sound levels recorded by the 
towed hydrophone with levels recorded by a hydro¬ 
phone when no wake is present in the direction of the 
noisemaker. Unfortunately, the levels recorded by 
the stationary hydrophone, suspended from the boat 
outside the wake, cannot be used, because the direc¬ 
tivity pattern of the noisemaker is unknown. It 
should be noted that the aspect of the noisemaker, as 
viewed from the towed hydrophone, is practically 
constant, while the aspect of the noisemaker relative 
to the stationary hydrophone changes by about 90 
degrees while the destroyer is moving toward, or re¬ 
ceding from the point of closest approach. 

The sound levels obtained by the towed hydro¬ 
phone with the noisemaker towed at a depth of 40 ft 
may serve as an approximate reference level repre¬ 
senting the wake-free state, because then most of the 
path from the noisemaker to the hydrophone runs 
below the wake. Subtracting these sound levels from 


the ones applying to the noisemaker towed in a wake, 
an approximate value for the transmission loss along 
the wake is found. The numerical values are about 
6 db for 3-kc sound and about 13 db for 8-kc sound 
at a speed of 10 knots; at 14 knots, the values are 
about 13 db and 30 db for 3-kc and 8-kc sound re¬ 
spectively. These transmission losses are of the same 
order of magnitude as those found in propagation 
across wakes. The increase of transmission loss with 
frequency is also in agreement with what has been 
learned about sound transmission across wakes. 

However, for the entire range covered (100 to 
1,000 ft) the transmission loss along the wake does 
not show the expected increase with distance from 
hydrophone to noisemaker. The sound levels used as 
reference values, with the noisemaker 40 ft below the 
surface, vary inversely as the square of the distance 
between hydrophone and noisemaker. But the sound 
levels recorded with the noisemaker in the wake also 
follow approximately the same inverse square law. In 
other words, the measured transmission anomalies 
fail to show any increase with distance behind the 
noisemaker, which would readily be interpreted as 
caused by attenuation inside the wake. There is even 
a slight decrease, perhaps 3 or 4 db, over a range of 
1,000 ft; however, this decrease may result from the 
presence of bottom-reflected sound. Measurements of 
the destroyer ship sound, with no noisemaker present, 
gave results similar to those obtained with the noise¬ 
maker. These observations are rather puzzling. 

The measurements of the sound output of a de¬ 
stroyer, cruiser, and aircraft carrier 2 give additional 
evidence of a very low transmission loss along wakes. 
During the so-called Z runs, the vessel to be measured 
passed the measuring vessel, which was hove to, and 
then made a turn so that, during the receding run, the 
axis of the wake coincided with the line connecting 
the stationary vessel with the receding one. The sound 
levels recorded were corrected for the transmission 
loss resulting merely from geometrical divergence ac¬ 
cording to the inverse square law, and from the cor¬ 
rected levels a transmission anomaly was derived. 
Attenuation coefficients along the wake were found 
to be 10 to 80 db per kyd; these attenuation coef¬ 
ficients were not judged sufficiently accurate to war¬ 
rant a discussion of variation with speed (10 to 30 
knots), frequency (5, 25, and GO kc) and ship type. 
In order to appreciate fully how small those attenua¬ 
tion coefficients measured along wakes are, it should 
be remembered that the measured transmission loss 
across wakes (see Section 32.3) corresponds to attenu- 



510 


OBSERVED TRANSMISSION THROUGH M AKES 



DEPTH IN FT 

Figure 3. Ten-inch propeller, 1,600 rpm. 

ation coefficients of 300 to 6,000 clb per kyd. This 
enormous difference is apparently real but has not 
yet been explained. 

32.5 TRANSMISSION LOSS IN MODEL 
PROPELLER WAKES 

Attenuation measurements on wakes of ships under 
way have been supplemented by experiments with 
wakes of a stationary model propeller. At the Woods 
Hole Oceanographic Institution, 2 an electrically oper¬ 
ated device was constructed for driving submarine 
propellers at speeds ranging from 266 to 1,600 revolu¬ 
tions per minute at various depths. This equipment 
was used in water 70 ft deep. 

First the relation between sound output and speed 
of the propellers at constant depth was studied, and 
the critical speed marking the onset of cavitation was 
determined. Four propellers ranging from 10 to 20 in. 
in diameter were employed. The noise level increased 
sharply whenever the tip speed of the propeller 
blades exceeded 33 ft per sec. In earlier experiments 
with 2-in. propellers, mounted in an experimental 
chamber, a critical speed of 35 ft per sec had been 
found at the same hydrostatic pressure. The agree¬ 
ment between these two figures appears quite satis¬ 
factory. 

Precise measurements of the attenuation were ob¬ 
tained by an arrangement in which the transducer 
and hydrophone were mounted on opposite sides of 
the wake on a pipe frame attached to the boom carry¬ 
ing the propeller and held rigid by wire stays. The 
instruments were 9 ft behind the hub of the propeller. 
In this way the axis of the wake was made to pass 



Figure 4. Fourteen-inch propeller, 1,600 rpm. 

between the transducer and the hydrophone, which 
were on opposite sides of it at a fixed distance of 6 ft. 
This arrangement had the advantage that it was 
easy to handle and could be used in deep water with 
complete assurance that the position of the instru¬ 
ments relative to the propeller would not change. 
However, it did not allow any variation of the dis¬ 
tance between the instruments and the propeller. 
Hence, it was impossible to determine the decay rate 
along the wake. 

With this arrangement measurements of sound at¬ 
tenuation were made systematically at different 
depths and with different frequencies. Each measure¬ 
ment of attenuation involved the observation of the 
response of the hydrophone under three conditions: 

(I) with oscillator on and the propeller at rest; (2) with 
the oscillator on and the propeller turning; (3) with 
oscillator off and the propeller turning. By suitable 
combination of these data it was possible to correct 
the observations for the noise produced by the pro¬ 
peller. Typical results for the different propellers are 
illustrated in Figures 3 and 4. 

First, it will be noted that the attenuation in¬ 
creases with frequency, being almost absent at 10 kc 
and rising steadily to 60 kc. This increase with fre¬ 
quency, at any fixed depth, is so steep that it defi¬ 
nitely exceeds the increase with frequency of the 
transmission loss through destroyer wakes, which is 
approximately proportional to the square root of the 
frequency [see Section 32.3.1, equations (10) and 

(II) ]. Second, at each frequency, the attenuation 
diminishes considerably with depth. This effect is 
more pronounced at the higher frequencies. Since the 
destroyer wakes have an average depth of 20 ft, and 
the transmission loss through them is a sort of aver- 

































TRANSMISSION LOSS IN MODEL PROPELLER WAKES 


511 


age over this entire range of depths, the second effect 
partially cancels the first one. Moreover, the bubble 
population found in a destroyer wake may be quite 
different from that in the wake of the stationary pro¬ 
peller (zero slip), and any variation of the relative 
abundance of bubbles of different sizes is likely to pro¬ 
duce frequency-dependent acoustic effects. Hence, 
the discrepancy noted above is not alarming. 

In the case of the 10-in. propeller (Figure 3), the 
attenuation falls to almost zero at a depth of 40 ft for 
all frequencies. This phenomenon is consistent with 
the results obtained in the model chamber mentioned. 
In the model experiments it was found that fewer 
nonpersistent cavities were formed at higher pres¬ 
sures. According to the mechanism of bubble forma¬ 
tion described in Section 27.1 higher pressure causes 
a more rapid collapse of the cavities formed, before 
there is time for a considerable amount of gas to 
diffuse into them; moreover, cavities containing a 
given amount of gas are compressed into bubbles of 
smaller size at higher pressures. 

Observations were also made at two frequencies 
with transducer and hydrophone in the wake, both 


mounted in the wake axis, but their number is small 
and no clear-cut conclusions can be drawn from them. 
There is some indication that for 50-kc sound the out¬ 
put of the transducer may be reduced, or “quenched” 
by the wake, but for 10 kc. the “quenching” effect, if 
it exists at all, is very much smaller than for 50 kc. 

In summary, the observations of wakes produced 
by a stationary propeller are in reasonable agreement 
with those of destroyer wakes, as far as the depend¬ 
ence on frequency of the transmission loss is con¬ 
cerned. By dividing the attenuations plotted in 
Figures 3 and 4 by the distance between transducer 
and hydrophone (6 ft), attenuation coefficients can be 
computed. For instance, at a depth of 15 ft attenu¬ 
ation coefficients of 3.6 and 2.3 db per yd are found 
for 25-kc sound, which is the same order of magnitude 
as found for destroyer wakes at speeds of 10 to 15 
knots. Since the diameter of the wake probably was 
smaller than the distance from transducer to hydro¬ 
phone, the values quoted for the attenuation coef¬ 
ficient are actually lower limits; the true attenuation 
coefficient may have been greater by 50 to 100 per 
cent. 



Chapter 33 


OBSERVATIONS OF WAKE ECHOES 


E choes from wakes, like those obtained from 
other targets, vary considerably with the type of 
sound gear employed, the prevailing oceanographic 
conditions, and the physical constitution of the wake. 
Before the observations are reviewed, it is necessary 
to outline the theoretical concepts entering into the 
reduction of the crude data obtained by measure¬ 
ment. 

As regards the physical mechanism by which sound 
is returned from a wake to an echo-ranging trans¬ 
ducer, two limiting cases can be imagined. On one 
hand, the multitude of microscopic scatterers may be 
spread out so thinly that the phases of the scattered 
sound waves are distributed at random — that is, so 
that constructive and destructive interference are 
equally probable. Then the average power returned 
to the transducer is obtained by summing up the 
contributions from the individual scatterers. On the 
other hand, a wake might reflect sound specularly. 
This alternative would occur only if the concentra¬ 
tion of scatterers near the wake surface increased in¬ 
wardly very rapidly. It is undecided as yet whether 
or not specular reflection from wakes does occur; in¬ 
conclusive evidence on this point will be discussed in 
Chapter 34. In the present chapter, wake echoes will 
be treated on the first assumption. Experience has 
shown that this approach is usually quite satisfactory. 

33.1 CONCEPT OF WAKE STRENGTH 

33.1.1 Target Strength of a Wake 
and Wake Strength 

Echoes from wakes differ in two important respects 
from echoes from ships and other small targets. The 
concept of target strength has been analyzed in Sec¬ 
tion 19.1 where it was shown that for a target of finite 
size the target strength becomes independent of range 
at very long ranges and may be computed from the 
equation 

T = E-S + 2H, (1) 


where E is the echo level in decibels above 1 dyne 
per sq cm, S the source level or pressure level 1 yd 
from the projector, in decibels above 1 dyne per sq 
cm, and // the one-way transmission loss from the 
source to the target in decibels. If equation (1) were 
used to compute the target strength T w for a wake 
from the echo level at long ranges, T w would increase 
with the range because, for practical purposes, the 
wake extends infinitely in the horizontal direction; as 
the range increases, more of the wake becomes ex¬ 
posed to the sound beam, more scattering occurs, and 
the target strength increases. For the same reasons, 
a transducer with a broad horizontal beam would 
yield a higher echo level than a transducer with a 
narrow pattern beaming sound at the same wake, 
other things being equal. 

It is desirable, therefore, to introduce in place of 
the target strength of a wake another characteristic, 
which is essentially the target strength of a 1 -yd 
length of the wake. This quantity is principally a 
function of the geometric dimensions and of the 
physical properties of the wake alone and, therefore, 
will be called wake strength and denoted by the sym¬ 
bol W. The wake strength will here be defined in a 
simple manner for an ideal wake, without regard to 
the physical structure of actual wakes. In Section 
33.1.2, an analysis of this wake strength in terms of 
the physical constitution of the wake will be given, 
including the effects originating from the finite length 
of the sound pulses used in practice. 

The wake echo will now be treated as if it were the 
echo from a plane strip having infinite horizontal ex¬ 
tension ( — oo<y<-j-oo) and a constant vertical 
height h (depth of the wake) which is supposed to be 
much smaller than the distance to the transducer. 
The hypothetical strip is assumed to have a rough 
surface, so that the reflection of sound by it is non- 
specular and perfectly diffuse, with a dimensionless 
coefficient of reflection s which is the fraction of sound 
energjr returned into a unit solid angle. The fraction 
of sound energy reflected back, regardless of direc- 


512 


CONCEPT OF WAKE STRENGTH 


513 


tion, is then 2v.\. By comparison with Section 19.1 it 
is readily verified that the target strength of one 
square yard of this wake surface, placed perpendicu¬ 
larly to the sound beam, is 10 log s. Since the depth 
of the wake is h yards, the target strength of a 1-vd 
length of wake is 10 log hs. In order to relate this 
wake strength to the observed echo intensity, con¬ 
sidering the directivity of the transducer and the 
scattering of sound from elements of the wake surface 
not perpendicular to the sound beam, a more detailed 
exposition of this simple case is required. 

The geometry of this experimental situation is 
illustrated in Figure 1, from which it is apparent that 


tional to the output voltage of the transducer acting 
as hydrophone. 

By substitution of the perpendicular range D from 
transducer to wake, 

D = r cos (/3 + 0) 
y = D tan (fi + <f>) , 

equation (3) is transformed into 

•t> = + \-e 

h = j 6(0)5 '(0) i0“°- 2raO 860 w+ * ) cos 3 03 + <f>) d4>, 
<*> = - £ -0 

or, to a very good approximation 


NORMAL TO THE WAKE TRANSDUCER AXIS WAKE 



Figure 1. Horizontal plane through transducer and 
wake. 

the surface element of the strip having the area hdy 
receives from the transducer the power 

10 -0.1ar 

Io - — T~~ 6(0) cos 03 + <t>)hdy, (2) 

r 2 

where 7 0 is the output of the transducer on the axis; 
6(0) measures the angular variation of the output 
around the horizontal plane; r is the distance from 
the transducer to the surface element hdy; 0 is the 
angle between this ray and the axis of the transducer, 
which subtends the angle /S with the normal to the 
wake so that cos (/3 + 0) measures the geometric 
foreshortening of the insonified area; and a is the 
coefficient of attenuation in the ocean. If the trans¬ 
mission loss on the return path is taken into account, 
the equivalent echo intensity I e is 

V — + 00 

r io“°- 2ar 

I e =sI 0 J ———6(0)6'(0) cos (0 + <t>)hdy , (3) 

y = - 00 

where 6(0)6'(0) is the composite pattern function of 
the echo-ranging transducer. The factor &'(<£) is the 
ratio between the response of the hydrophone to a 
signal incident at an angle <j> to the axis and the 
response to a signal of equal strength incident on the 
axis. Thus the equivalent echo intensity 7 P is propor¬ 


&O)6'(0) cos 3 (/3 + <f>)d(t> ■ (4) 

This approximation neglects the variation of the 
transmission anomaly along the wake, which is in¬ 
significant because of the narrow beam pattern of the 
transducers used in practice. By writing- 

cos (/3 + <j>) = cos /3 (cos <f> — tan /3 sin 0) 
and D cos (3 = f, 


_f jQ0.2aDsec^2)3 

1 0 


= hs f 


where r is the range to the wake measured along the 
transducer axis, equation (4) becomes 


0 =+ s -0 


— 10 °- 2ar r 3 = hs f 6(0)6'(0)(cos 0 — tan/3 

7n J 


sin 0) 3 d0. 

(5) 


By collecting the terms representing the transmission 
loss, 

ar -j- 20 log r = 77, (6) 


and adopting the abbreviation 

*=+l 

10 log / 6(0)6'(0)(cos 0 — tan /3 sin 0) 3 d0 = 4 / , (7) 


equation (5) can be expressed, in decibels, as 
E — S + 277 — 10 log r — T = 10 log hs = W. (8) 


The quantity 'k defined in equation (7) will be called 
the wake index, analogous to the reverberation index 
defined in Chapters 11 through 17. The product hs 
in equation (8) has the dimension of a length. Since 
the ranges appearing in (8) are customarily measured 
in yards, the wake strength W is the ratio of hs to one 
yard, expressed in decibels. Then, by comparing equa¬ 
tions (8) and (11), the relation between the wake 








514 


OBSERVATIONS OF WAKE ECHOES 


strength W and the target strength of a wake T w 
becomes 

T w = W + 10 log f + T, (9) 

where f is the range to the wake, measured along the 
transducer axis, and 4/ is the wake index defined by 
equation (7). The physical meaning of equation (9) 
has already been noted in the first paragraph of this 
section. 

33.1.2 Dependence of Wake Strength 
on Physical Parameters 

The fundamental definition of wake strength, 
given in the preceding section, was facilitated by 
treating the wake as if it were a plane strip with the 
coefficient of reflection s. In effect, this approach 
neglected the wake structure along the transverse x 
axis. Moreover, that analysis tacitly assumed the 
use of a continuous signal for measuring the wake 
strength. But if short sound pulses are beamed at a 
diffusely reflecting plane, the echo profile on the 
oscillogram, in general, will not reproduce the shape 
(usually square-topped) of the signal, and the de¬ 
pendence of echo intensity on signal length must be 
investigated. For a brief theoretical demonstration of 
this fact see Section 19.3. 

On inspection of the sample oscillograms repro¬ 
duced in Figure 7 of Chapter 30, it will be observed 
that reflection from a wake also alters the shape of 
square-topped sound pulses. However, the explana¬ 
tion of this effect is more complicated than that of the 
variation with pulse length of the echo intensity re¬ 
turned by a plane target. In fact, the echo profile de¬ 
pends on the transverse structure of the wake, which 
therefore must form an integral part of a compre¬ 
hensive theory of wake echoes. 

According to the working hypothesis adopted in 
Section 26.3 wake echoes are composed of a multitude 
of reflections originating throughout the entire wake. 
Superposition of these scattered waves leads to con¬ 
structive and destructive interference, because their 
phases are distributed at random. Consequently the 
echo intensity measured at any instant will not equal 
the average value. The difference between the in¬ 
stantaneous and average echo intensity is a rapidly 
fluctuating quantity, evidently beyond the reach of 
theoretical analysis, because it depends on the micro- 
structure of the entire wake. Physical significance can 
be attributed only to the average of many echo pro¬ 
files recorded in rapid succession. Such averaging is 
also necessary in order to minimize the effect upon 


the echoes of the rapid fluctuations of the transmis¬ 
sion loss in the ocean at large, which were discussed in 
Chapter 7. Accordingly, the theoretical analysis 
about to be presented refers to “average” echo in¬ 
tensities throughout. 

The problem now is to evaluate the relation be¬ 
tween the total number, arrangement and physical 
parameters of the bubbles and the overall reflectivity 
s of the wake, filling a volume of constant depth h 
and width w and of infinite length ( — °°<y<+°°). 
Let n(x) be the number of bubbles per unit volume 
at the distance x from the nearest boundary of the 
wake (0 < x < w); n(x ) is supposed not to vary 
appreciably along the wake axis over a distance of 
the order of the width of the sound beam, or dn/dy 
« dn/dx. With a s and a, representing the cross 
sections for scattering and extinction defined by 
equations (34) and (43) of Chapter 28, the echo re¬ 
turned by an individual bubble has the intensity 

I e = — I, —6(<A)6'(0) e~ 2aeN{x) sec (3 + *’. (10) 

47r r 4 

The fraction oj 4tt of the incoming sound energy is 
scattered into the unit solid angle, and the term 
IQ-o . 2 arjfi re p resen t s the two-way transmission loss in 
the ocean. The sound beam is trained obliquely at the 
wake, so that the axis of the transducer and the nor¬ 
mal to the wake include the angle /3- It is the oblique 
path of the sound beam traversing the wake which 
accounts for the factor sec (/3 + cj>) in the exponent 
expressing the two-way transmission loss inside the 
wake, which is based on equation (52) of Chapter 
28. The geometry of the situation is illustrated in 
Figure 2. The echo returned by a volume element of 
the wake is found by multiplying equation (10) by 
n(x)hrdrd<t> 

dl e = k ~ h 10 v - b{4>)b'{<j>)n{x)e~ 2aeN{x) sec {0 ^ ] drd4> 
47r r 

( 11 ) 

on the assumption that all bubbles have the same 
size, so that the cross sections cr s and a e are constant 
throughout the wake. Finally, the echo returned by 
the entire wake can be evaluated by integration be¬ 
tween the appropriate boundaries, which must be 
chosen carefully; these boundaries are essentially de¬ 
termined by the width of the wake and by the signal 
length. 

When an echo-ranging signal is sent out into the 
water, the volume from which echoes can be received 
at anyone time fills a spherical shell of thickness cr/2, 
where r is the duration of the square-topped signal 





CONCEPT OF WAKE STRENGTH 


515 


and c is the sound velocity. This region, which travels 
outward at a velocity c/2, may for purposes of dis¬ 
cussion be referred to as the volume occupied by the 
echo-ranging signal or pulse. As the pulse travels out¬ 
ward, it will cross the wake, which has roughly the 
shape of a cylinder of infinite length. The determina¬ 
tion of the echo strength requires integration of equa¬ 
tion (11) over the volume in which pulse and wake 
overlap. In the general case, this volume has a rather 
complicated shape which, moreover, varies with time 
as the pulse travels across the wake. It should be 
noted that in most practical situations the depth of 
the wake is small compared with the range to the 
transducer, so that the vertical curvature of the 
pulse boundaries can be neglected; in other words, the 
pulse will then be treated rather as a cylindrical shell, 
with the cylinder axis normal to the ocean surface. 



Figure 2. Sound striking wake. 


Now there are two simple limiting cases for which 
the shape of the volume contributing to the echo can 
readily be visualized, namely when the signal length 
is either much greater or much smaller than the width 
of the wake. In the first case, the entire volume of the 
wake will contribute to the echo for a considerable 
length of time during which the average echo intensity 
will be constant. In the second case, the echo will 
come only from a thin spherical shell “cut out” of the 
wake, so that the average echo intensity will vary, 
while the pulse traverses the wake, without ever at¬ 
taining a constant value; the mode of this variation 
is a function of the density distribution n{x) across 
the wake. 

Long Pulses 

The case of very long pulses will be taken up first. 
If the signal length r 0 , which equals cr/2, is much 


greater than what might be called the slant width w' 
of the wake, which equals w sec (/? + </>), then the 
entire volume of the wake will be intersected by the 
pulse during a finite interval of time. During this 
period, the average echo intensity is constant, and 
can be found by integrating equation (11) over the 
wake volume. The limits of this integration are most 
readily given if the variable x is substituted for r, 
since then x varies between 0 and w, while <t> varies 
from ( — 7 t/ 2)— /? to (+ir/2)—/? (s?e Figure 2). By 
writing 

r = (r cos d - ~ + x'j sec (/? + </>), 

and r = (^D + ^ sec /?, (12) 

then dr = sec (/3 (f>)dx , 

and the new constant f is the range, measured on the 
axis of the transducer, of what might be called mid¬ 
wake — the point of intersection between transducer 
axis and wake axis. As in the preceding section, the 
variation along the wake of the transmission anomaly 
in the ocean will be neglected by setting 

i a- 0.‘2ar i/a— 0.2ar 


By substituting equation (12) in equation (11), the 
integral now reads 


-f» 10°' 2,I? 

/ 0 


ha s 
47r 


f h )b '(<t >) cos- (/? + <{>) see 3 /?' 

dxd<p. 


n (x)e ~ 2aeNi ' x ^ sec ^ + ^ 


1 + 


2x 
2 V 


i - w y 
P cos /?/ 


(13) 


Unfortunately, it is impossible to integrate this ex¬ 
pression in closed form. An approximation sufficient 
for all practical purposes will be given. 

Consider first the integral over dx, namely, 

2x ~ w ' n (x)e~ 2aeN{T>acc{0 + ,t> dx, (14) 


C w / 2x — w \ 
Jo \ " r 2F cos /?/ 


and apply the theorem of the mean value of an 
integral a to the inverse cube term in brackets; in 


a This well-known theorem states 


with 


.T2 X2 

f J{x)g{x)dx =f(x) f g(x)dx, 

Xl XI 


Xi< X < x 2 , 


as long as /(x) and g(x) are continuous over the range of 
integration, and g{x) is not negative. 
















516 


OBSERVATIONS OF WAKE ECHOES 


other words, put this term in front of the integral, re¬ 
placing x by an unspecified, constant, mean value x. 
With this procedure the integral over dx in equation 
(14) can be evaluated without further approxima¬ 
tion: 


i + —-—1 f 

2r cos J J o 


n{x)e~ 2aeN(x) dx 


1 + 


2x — u-1- 3 
2/\cos /3_l 

—2aeN(ic) sec (0-r<t>) COs(g + 0) 

2a c 


(15) 


Since, by definition, x is confined to the interval be¬ 
tween 0 and w, the inverse cube correction factor al¬ 
ways lies between the following limits: 


1 +- 


2 f J 


< 


1 +; 


2x — 


1-* r 

w sec 0 


L -- 

J L 

2 r J 


Since w sec /3 2 r is much less than 1, in most practical 
echo-ranging situations this correction factor is un¬ 
important, even at short ranges. In view of the 
limited accuracy obtainable with the current tech¬ 
niques of measuring echo levels, this correction factor 
will be omitted. The echo integral, equation (13), 
thus assumes the form 


1 h C 

— r®10°‘* af = I b(<t>)b'(<j>) [cos <t> — tan /3 sin 0] 3 - 

Iq 8ttct, J t 

- 2 -0 

[1 - e -2°'-V(v’)"*<e++)y 4> (16 ) 

Note that 

10 log e 2aeX(w) = 2H w 

is the two-way transmission loss in db for perpendic¬ 
ular transit through the wake, according to equation 
(8) of Chapter 32. 

It will be observed that for a high acoustic opacity 
of the wake [a e N(w) ^>1 ] the exponential in the 
bracket under the integral is very much less than 1. 
Hence, in the case of infinite acoustic thickness of the 
wake, there follows the rigorous formula 


+ - 


r 3 10° 


he 

8ira e 


— I b(<f>)b'\<t >)[cos $ — tan/3sin0] 3 d<£- 
rcr, J x 

2 ^ (17) 

By comparison with equation (5), the value of the 
wake index T ro , where the subscript has been added 
to emphasize that this index refers to infinite acoustic 
thickness, can be identified: 

+ \-t> 

'ken = 10 log J* b(4>)b'(<j>)[e os <j> — tan /3 sin 0] 3 d</>. (18) 


Conversely, for highly transparent wakes [ovV(u') 
« 1] the second bracket under the integral in equa¬ 
tion (16) may be developed into a series and the quad¬ 
ratic and higher terms neglected, yielding 

1 _ e -2«.w«o «*(* + *) = 2a,X(w) sec (0 + 0)- 

The formula for the wake strength of highly trans¬ 
parent wakes then reads 

— f 3 10°' 2ar = —- N(w) sec /3 f b(4>)b'(<f>) ■ 

1(1 47T Jr 

-2-0 

[cos 4> — tan 0 sin 4>yd<t>, 
'ko = 10 log | sec 0 f b(<t>)b'(<p) [cos <f> 

J-\-& 

— tan 0 sin 0] 2 d</> J • (19) 

Numerically, the difference between 'ko, in equation 
(18) and 'k 0 in equation (19) is negligible for direc¬ 
tional transducers as long as 0 is small. To a very 
good approximation the general equation (16) can, 
therefore, be written as 

7 f 3 10° 2 “ ? " (tt) = ~ [1 - e~ 2atmw) * * ] • 1 (20) 

/ o 07 T(J e 

Expressed on a decibel scale, equation (20) becomes 
E — S + 2H + 10 log r — 

- 10 log j — [1 - e ~ 2aeS(w) «* ■*] J- . (21) 

‘ 8ira e J 

Hence, the reflectivity of the wake per unit solid 
angle is 

s = — • - [1 - e ~ 2aeXUr '> ■* . (22) 

8tt a e 

By this equation, the problem proposed at the outset 
of this section is solved for sufficiently long pulses 
(r 0 > w). However, it should be remembered that 
equation (21) does not represent the entire echo pro¬ 
file, but applies merely to its central part which has 
a constant average intensity because the pulse over¬ 
laps the entire wake. The rise and fall of the average 
echo profile, when only part of the pulse intersects the 
wake, cannot be represented by a simple formula, be¬ 
cause of mathematical difficulties of the same nature 
as will become apparent presently in the discussion 
of short-pulse echoes. 

Short Pulses 

For pulse lengths smaller than the wake width 
(r 0 < w), equation (11) has to be integrated over the 

















CONCEPT OF WARE STRENGTH 


517 


volume in which the wake and the cylindrical shell 
(thickness r n , inner radius r x ) of the pulse intersect. 
Hence, 


I e 


<t>h + ro 



b(4>)b'(4>) ■ 


n(x)e~ 2aeX(x) sec (fi + ♦> drd<t> • (23) 
The limits <f> n and 4>i, of the integral over <!<$>, unspeci¬ 
fied tor the time being, are determined by the relative 
position of pulse and wake and, therefore, vary with 
time; their explicit form will be evaluated after the 
discussion of the integral over dr has been finished. 
Equation (23) is valid only for rectangular pulses —- 
that is, for echo-ranging signals whose intensity is 
constant for their duration. 

The variation of n(x) and N(x) across the wake 
prevents integration of (23) in closed form. In order 
to gain any insight at all into the behavior of short- 
pulse echoes, a drastic simplification becomes neces¬ 
sary. For this reason the discussion is confined to a 
wake of constant bubble density in the transverse 
direction. Putting thus 


and 


n(x) = constant = n = N(w)/w 
-Y(x) = nx 


the integral reads 


Ijl 

h 


<t>h r, ra 

hor. r no~°- 2nr 

Ai r 


If 


b(<t>)b'(</>)ne 


- 2 trenx sec (/3 + <t>) 


drdxp ■ 


(24) 


While in the general case <£>„ anti </>/, vary as r increases 
from r x to r x + r 0 , for short pulses this change is quite 
negligible. The integration over dr may then be 
carried out before the integration over d<f> without 
difficulty. On account of the geometric relation, from 
Figure 2, 

x sec (/3 + 4>) — r — D sec (0 + <j >), 


the integral over dr in equation (24) becomes 


r, + ro 

C 10 _ °' 2ar 

J r® ' 


[r — D sec (0 -f 4>)~\dr • (25) 


After applying the theorem of the mean value of an 
integral with respect to the factor 10“" 2 " r r 1 the inte¬ 
gral (25) is transformed into 

r*~ 3 10~°' 2ar *—— n g — 2<renro-j^-2aenCri—D sec (/? + $)] 

2 o,. 


Since r* differs very little from r x (because r x < r* 
< >’i + r 0 , by definition), r* can be replaced by r x 


without any appreciable loss of accuracy. Thus the 
echo integral, equation (24), reads 

<f>b 

f e r?10° 2ori = ^-[l - e - 2 --»] f b(<f>) . 

I o 07 T<T e J 

<i>* 

b'( 0 )e _2,7e " Cr ‘~ Osec W + 0)] d<f>. (26) 

The exponential under the integral measures the 
transmission loss resulting from absorption and scat¬ 
tering inside the wake, since r x — D sec (j3 + </>) is the 
distance, along any ray ( <f> = constant), from the 
inner boundary of the pulse to the front of the wake. 
Now by making the substitution 

?’i — I) sec (0 + <f>) 

= n — D sec 0 — D [sec (/3 -f <t>) — sec 0], 
equation (26) assumes the form 

<t>b 

— r\ 10°- 2 " n = - ><J - [ | _ e -2aennj e -2aen(r,-Daec0) . 

I o 87 ra c J 

<t>a 

V(<t>)e 2aenD Csec (3 + 0)_ sec n d<t>. (27) 

Here the factor (1 — e~- ,jenr °) comprises the effect of 
the pulse length on the echo strength. The transmis¬ 
sion loss inside the wake has been split into two 
factors. The first one, namely e -‘ 2 . cen( ~ ri - Dsecii ^ d e _ 
pends only on the range r x of the pulse, which in¬ 
creases with time, but not on the directivity of the 
transducer; in fact, this first factor is simply the 
transmission loss, measured along the sound beam 
axis, from the boundary of the wake to the pulse. The 
second factor is independent of time and appears as 
an exponential under the integral over d<j>. Using the 
abbreviation 

<t>b 

101 og|jl(<^) 6 , (^) 10 2<re ” £>t:sec ( . a +^- 8ec «^|, (28) 

<t>« 

which defines another wake index applying to short- 
pulse echoes, equation (27) may be written on a 
decibel scale as follows. 

E - S + 2H + 10 log r x - 

= 10 log [1 - e^roy^aeMn-Dtecfi) l (2g) 

vStto” e ) 

The quantity (r x — D sec 0) is the distance, meas¬ 
ured along the transducer axis, which the rear 
boundary of the pulse has penetrated into the wake. 
Hence, for a directional transducer no appreciable 
echo intensity will be obtained outside the range of 
penetration which is given by 

D sec 0 < ?t < (D + iv ) sec 0. 






518 


OBSERVATIONS OF WAKE ECHOES 


During the time interval in which r, is confined be¬ 
tween the limits stated, equation (27) represents the 
average profile of short-pulse echoes. Consequently 
the average echo intensity falls off exponentially from 
its maximum value, attained immediately after the 
pulse has fully entered the wake (rj = D sec /3), pro¬ 
vided that the value of the integral over r/</> does not 
vary with time. This condition can indeed be realized 
in special cases. 



Figure 3. Successive positions of ping inside wake. 


Although the integrand in equations (27) and (28) 
is independent of time, the range of integration is not, 
in the general case. This fact is illustrated by Figure 
3, showing successive positions of a short pulse 
beamed obliquely at the wake. As the ptdse crosses 
the wake, the limits, 4> b and 4> a of the integral in equa¬ 
tion (28) increase steadily. Thus the wake index is, in 
principle, a variable quantity, which would seem to 
impair the usefulness of this concept. However, the 
directivity of the transducer effectively limits the 
angular width of the sound beam. If it were possible 
to place the transducer in such a position that the ef¬ 
fective half-width of the sound beam remained 
smaller at all times than the boundary 4> b , the varia¬ 
bility of <pb and 4><i would become irrelevant for all 
practical purposes, and the wake index 'k' would 
actually be a constant. 

In order to formulate this idea quantitatively, note 
that the boundaries of integration in the wake index 
are explicitly 



These formulas can readily be verified by inspecting 
Figure 3. If the effective angular width of the sound 
beam be called 2</>', so that the effective half-width is 
<t >', the conditions under which 4>' becomes constant is 

4>' ^ <t>h • (31) 


By substitution of equation (30) in equation (31), it 
follows that 


-V\ + r 0 J 


' + P, 


(32) 


1) 


i'i +r c 


^ cos (0 -f- <//)• 


Write 


T\ = D - f- 0*i , 

where, according to Figure 3, X\ is confined between 
the following limits 

0 < x\ < w , (33) 

and substitute in equation (32); then 


1 + (an + r 0 )/D " 

or approximately, since generally (xi + r (l )/ 1) is much 
smaller than one, 

1 - t ? ° ^ COS (/J + <£')• (34) 


I) 


Hence, 


D £ 


Xi + r 0 


1 - cos 03 + </>') 


This inequality may be put into a more stringent 
form by setting xi equal to zero, on account of equa¬ 
tion (33), so that the desired condition takes the 
final form 


D <C 


(35) 


1 - cos QS + <t>') 
which assumes that right from the moment the pulse 
has entered the wake, or Xi = 0, the effective half¬ 
width of the sound beam is smaller than the variable 
boundary <fi b . Any less stringent form of the condi¬ 
tion, such as x\ ^ w, would distort the representa¬ 
tion of the entire echo profile indicated by equation 


(29). 

For numerical evaluation of equation (35), <p' = 0 
degrees appears to be a reasonable value for trans¬ 
ducers of conventional design used in echo ranging. 
Results for two typical cases are given in Table 1. 
Since the shortest signals used in practice correspond 
to r 0 - 1 msec - 0.8 yd, condition (35) is easy to 
maintain in ranging perpendicularly at the wake. 
Accordingly, the upper limit 4> b in equation (28) may 
be replaced by a practically constant value 4>' if the 
range D from the transducer to the nearest point of 
the wake is chosen so as to comply with the first case 
in Table 1. 

On the other hand, it is evidently impossible to 
satisfy the second condition in Table 1 — in other 






















CONCEPT OF WAKE STRENGTH 


519 


words, with markedly oblique incidence of the sound 
beam the concept of wake loses its usefulness for 
short signals. The theoretical derivation of the echo 
profile for short-pulse echoes obtained with an 
obliquely trained transducer would require extensive 
numerical integrations of equation (27), taking into 
account the continual variations of the boundaries 
4> a and <pb- 

Table 1 . Conditions for constant wake index. 


Effective half-width of sound beam <f>' = 6° 


I Sound beam trained 

Condition 135) 

perpendicularly at 


wake 0 = 0° 

D < 200n, 

II Sound beam trained 

Condition (35) 

obliquely at wake 


0 = 60° 

D < 1.66r„ 


Summing up the discussion of short-pulse echoes, it 
should be remembered that the analysis, for mathe¬ 
matical reasons, had to be restricted to wakes having 
a constant bubble density in the transverse direction. 
According to the varying transverse structure of 
actual wakes, their echo profile will deviate somewhat 
from the exponential shape given by equation (27), 
even if the sound beam is trained perpendicularly at 
the wake. 

33.1.3 Definition of Wake Strength 

The concept of wake strength has been introduced 
in Section 33.1.1 in the hope of arriving at a wake 
characteristic that is a function solely of the geomet¬ 
ric dimensions and physical parameters of the wake. 
But the detailed analysis in Section 33.1.2 showed 
that this aim defies complete realization. The strength 
of wake echoes depends on the signal length and on 
the directivity pattern of the transducer in a rather 
complicated manner; this dependence cannot be for¬ 
mulated mathematically in a simple way without in¬ 
troducing various approximations. These effects are 
small compared with those resulting from the varia¬ 
bility of echo strength with range; this large variation 
with range can be eliminated from the wake strength 
by a suitable definition of this quantity. 

We now define wake strength by the equation 

W = E - S + 2H - 10 log r - * , (36) 

where E is the echo level, S the source level, H the 
one-way transmission loss from the transducer to the 
wake, r the range in yards of the wake, and T is the 
appropriate wake index. This definition comprises 


both equations (21) and (29), referring to long and 
short signals, respectively, and implicitly disregards 
the small difference between the ranges defined as r 
and ri. For all practical purposes the range to be used 
in the computation of wake strength may be de¬ 
termined from the time interval, purposely recorded 
on the oscillogram, between midsignal and the in¬ 
stant to which the measured echo amplitude refers. 
When the difference between the transmission loss H 
and the inverse-square loss 20 log r increases linearly 
with range, an alternative way of writing equation 
(36) is 

W = E — S + 2 ar — 10 log m 2 + 30 log r — F, (37) 

where a is the coefficient of attenuation in the ocean 
expressed in db per kyd, and (m — 1 ) is the reflec¬ 
tivity of the ocean surface, so that the factor y? repre¬ 
sents the increase of echo intensity caused by the 
double path resulting from surface reflection. To be 
consistent with the standard procedure for comput¬ 
ing target strengths, p should be put equal to 1 (see 
Section 22.2), so that equation (36) reduces to b 

W = E - S + 2 ar + 30 log r - (38) 

Finally, according to equation (9), the target strength 
of the wake T w is given by 

T w = W + 10 log r + *. (39) 

The wake index F was first defined by equation 
(7), on the assumption that reflection from a wake 
can be treated like that from a plane strip. This is, 
indeed, a good approximation provided that the wake 
is highly opaque, or N(w) » 1, and long signals are 
used, or r 0 w (for example, r 0 > 2w), according to 
equation (18) which turned out to be identical with 
equation (7): 



^ 00 = 10 log I 6(<£)&'(<£) [cos $ — tan/3 sin 03 3 d<£. (40) 


b All the numerical values of W reported in this chapter 
have been computed according to the definition given by 
equation (38). However, when the original publications are 
consulted, care should be taken in ascertaining what particular 
definition of wake strength was used by the author. For in¬ 
stance, in one paper, 1 n is set equal to 2 in correcting echo 
levels for transmission loss; moreover, a term 10 log (4-7r) is 
also added to equation (37). The net result is that the values 
of the wake strength in reference 1 are 5.0 db larger than those 
computed from equation (38). 

Since the reflectivity of a target has been defined, in Section 
19.1 of this volume, in terms of the sound reflected into a unit 
solid angle, it seems desirable to maintain the same convention 
for the reflectivity s of a wake. Accordingly, the term —10 log 
(47 t) here appears in equations (45) and (46), instead of in 
equation (37). 










520 


OBSERVATIONS OF WAKE ECHOES 


However, if the wake is highly transparent, or 
A (w) « 1, the echo level E, and also W as computed 
from 'Per, will be found to increase with the oblique¬ 
ness 0 of the impinging sound beam. In this case, the 
replacement of 'Fa, by 'ko, according to equation (19), 
may be expected to give a wake strength independent 
of 0: 


’ 2 ^ -> 

'kn = 10 log |seci3j b{<t>)b'(4>) [cos<£ — tan/3sin«^] 2 d<^>|. 


(41) 


For short pulses (r 0 « w, say r 0 <w/2) the appro- 
priate value of 'k is 
+ </>’ 

= 10 log | j e 2aenr (se< ' ♦ “ 1} b(<t>)b'(<t>)d<t> j . (42) 

— 0' 

This formula is a specialization of equation (28) for 
/3 — 0; hence D in equation (28) is approximately 
equal to r. Moreover, according to equation (35), 
the range from transducer to wake must be less than 
200 times the pulse length, if equation (42) is to be 
valid. For a sound beam trained obliquely at the 
wake — or for 0 ^ 0 — equation (28) loses its useful¬ 
ness, as the wake index becomes a quantity varying 
with time. 

Though their mathematical expressions appear 
rather different, the numerical discrepancy between 
'k' and 'k 00 is quite small, probably never exceeding 
1 db for transducers of high directivity. For signals of 
intermediate length, of the order of the wake width, 
the mathematical analysis is difficult; it is suggested 
simply that 'k co be used. For practical purposes, the 
differences between the various types of wake indices 
might be neglected altogether, by writing for perpen¬ 
dicular incidence of the sound beam 


'k = 10 log J* b(<t>)b'(<l>)d<t >. (43) 

This approximate formula reveals a close relationship 
between t he wake index and t he surface reverberation 
index, defined in a University of California Division 
of War Research [UCDWRJ report, 2 as 


Hence, 


J, = 10 log 


2i r 


/« 


b{(f>)b\<t>)d4> 


J s -f- 8 db — 'k. 


(44) 


Furthermore, it is of interest to compare the for¬ 
mulas giving the wake strength as a function of the 
physical parameters and of the pulse length: 


Short pulses, r 0 « w 
W = 10 log hs 

j h a. 


- 10 log 


(87T a, 


1 


*] 


2<renro I —2 cen^ri — D sec /3) 




(45) 


Long pulses, r 0 » w 

W = 10 log hs = 10 log ! — -|~1 - e ~ 2aeN(w) (46) 
I 877(7,1 ) 


It will be noted that the first exponential in equation 
(45) becomes equal to that in equation (46) for r 0 = ic, 
because of equation (54) of Chapter 28, reading 
hw — N(w), 

which is the definition of n. Equation (45) is an ap¬ 
proximate formula, because in its derivation the as¬ 
sumption n(x) = constant = h had to be made. 
However, while equation (46) applies to the constant 
average echo intensity constituting the central part of 
long-pulse echoes, equation (45) represents the entire 
average profile of short-pulse echoes. 

So far the entire discussion has been restricted to 
average echo intensities. But, as described in Section 
30.1.3, peak echo intensities are customarily measured 
by the San Diego observers. The measurements re¬ 
ported in Chapters 11 through 17 of this volume on 
the “band” or “point” method of reading reverbera¬ 
tion records imply that about 6 db must be sub¬ 
tracted from the wake strengths computed by equa¬ 
tion (38) from peak intensities, in order to express 
them on the scale of average intensities envisaged in 
equations (45) and (46). This correction will be ap¬ 
plied only in Section 34.3.1, where the interpretation 
of the observed wake strengths by the acoustic theory 
of bubbles is discussed. In that context, the wake 
strength computed from the measured peak ampli¬ 
tudes of short-pulse echoes, and then corrected by 
subtracting 6 db, will be regarded as corresponding 
to the maximum of the profile (45), or 
ri — D sec 0 = 0. 


33.1.4 Decay Kate of Wake Strength 


The decay rate of wake strength, in terms of the 
physical properties of the wake, is found by differen¬ 
tiating equations (45) and (46) with respect to the 
time. Before doing so, it is advantageous to make the 
substitutions 


and 


H w 

4.34<r e tc 


A» = - 


H w 


4.34(7, 






CONCEPT OF WAKE STRENGTH 


521 


where H u , is the one-way transmission loss for hori¬ 
zontal passage of sound through the wake, as defined 
in equation (8) of Chapter 32. Equations (45) and (46) 
then read 

IF = 10 log i ~ -~[1 — e “°- 46ff " ro/ ”’]l (r 0 « w) (47) 

t 07T <T e ) 

w = 10 log | — -^[1 - e -°' 46 ""’] j (r 0 » w). (48) 
The result of the differentiation is: 


Short pulses, r 0 « w 


dW o 1 dh 0.46e~ O4fi// “ ro/u ' r 0 
dt ~ 4 34 h dt + 1 — e -° A6H '‘ r <‘ /w w ' 

P dH w H I,, dw 

L dt w dt _ 

Long pulses, r 0 w 

dW _ 1 dh 0.46e~° 46// "’ dH w 

dt ~ 4 34 h dt + 1 - e -°- 46// » * ~dt" 


(49) 


(50) 


The first term in these equations is the same for long 
and short signals. It represents the effect of the 
change in depth of the wake and is known to be quite 
small; for two destroyer wakes, according to data re¬ 
ported in Section 31.3.1, (1 /h) (dh/dt) was found to be 
— 0.08 and +0.04 db per minute, respectively. The 
second term seems to be the dominant one. While the 
factor in front of it, containing the exponential, is ex¬ 
ceedingly small for fresh wakes, it grows rapidly and 
approaches infinity for very old wakes; numerical 
values of this factor can be read from the graph in 
Figure 4. 

The differential quotient dH u /dt is equal to 3 db per 
minute, for destroyer wakes, and ( H w /w ) ( dw/dt ) can 
be estimated from the same data to be of the order of 
1 to 2 db per minute. It will be noted that equation 
(49) would be transformed into equation (50) by 
setting r 0 /w equal to one — except for the term pro¬ 
portional to (H, v /w) (dw/dt) which does not appear in 
equation (50). The physical meaning of this term is 
interesting: as the wake ages, it spreads laterally, 
causing dw/dt to be a positive quantity; conse¬ 
quently, the factor H w /w is bound to decrease, even 
if H w , the total attenuation across the wake, remains 
constant. According to equations (46) and (48), for 
long pulses the wake strength is a function of N(w), 
which is directly proportional to H,„ or the total atten¬ 
uation, and which is not affected by a mere spreading 
laterally of the wake without simultaneous disinte¬ 
gration of the bubble population. But for short pulses 



H, IN 08 

Figure 4. Factor appearing in formula (50) for the de¬ 
cay rate of wake strength. 

[see equations (45) and (47)] the wake strength is a 
function of the product of signal length r 0 times the 
average bubble density n which is proportional to the 
attenuation coefficient. Hence, the decay rate of the 
wake strength for short pulses is a function of the 
decay rate d(H«./w)/dt of the attenuation coef¬ 
ficient. which gives origin to a term proportional to 
(dw/dt) or the lateral spreading, even if dH w /dt is 
negligibly small, corresponding to an extremely small 
physical disintegration of the bubble population. 

Summing up, for short pulses, whose volumes do 
not intersect the entire wake, there exists a progressive 
decay of wake strength having a purely geometric 
origin — namely the lateral spreading of the wake. 
Naturally, for long pulses, which overlap the entire 
wake, such an effect cannot arise. If the decay of a 
wake is followed over a very long period of time, and 
a constant pulse length is employed, it may well 
happen that the pulse which was chosen, at zero age 
of the wake, so as to be long will finally become short 
with respect to the steadily growing width of the 
wake. At the moment the critical point w — r 0 is 
passed, the term ( H w /w ) (dw/dt) suddenly begins to 
operate, causing an accelerated decay. In order to 
avoid all unnecessary complications, it may be ad¬ 
visable, therefore, to choose very long signals for the 
study of the decay rate of wake strength. 

The general significance of equations (49) and (50) 
is that they establish a relation between the decay 
rates of the transmission loss and the wake strength 
which can be tested by observation. 





















522 


OBSERVATIONS OF WAKE ECHOES 


33.2 EXPERIMENTAL PARAMETERS 

The formulas derived in the preceding section are 
applied in later sections to the interpretation of echoes 
from actual wakes. The theoretical results may also 
be applied to indicate what type of echo-ranging ex¬ 
periment is most suited for fundamental studies of 
wakes. Certain considerations along this line, espe¬ 
cially concerning the choice of transducer directivity, 
pulse length, and frequency are presented in this 
section. 

33.2.1 Transducer Directivity 

The mathematical intricacies of the analysis given 
in Section 33.1.2 are essentially a consequence of the 
imperfect directivity of the transducers and the finite 
range over which the wake is observed. The chief re¬ 
sult is a variety of wake indices pertaining to specific 
experimental situations. Fortunately, the picture is 
greatly simplified in practice, because of the proper¬ 
ties of the transducers customarily employed in echo 
ranging. 

The numerical differences between the different 
wake indices, for the same directivity pattern, are 
quite insignificant in proportion to the accuracy at¬ 
tainable in acoustic measurements. As an example, 
Table 2 gives the wake indices computed from 
the composite directivity pattern of a particular 
transducer. 

The integral 'k taken over the composite directivity 
pattern alone, as defined by equation (43), is given 
for comparison. It would seem that F = J s + 8 db 

Table 2. Typical wake indices — UCDWR trans¬ 
ducer No. 1917 at 45 kc. 

/3 = 0° d = 60° 

'Foo -9.75 db -9.87 db 

*0 —9.75 —6.85 


2 <7 e nr = 20 —9.58 

2 (T e nr = 40 —9.52 . . . 

* -9.74 -6.74 


may be used in place of any of the rigorous values of 
the wake indices, except for Fi, with obliquely imping¬ 
ing sound beam (/3 = 60 degrees). In this particular 
case the wake index includes the factor sec /3, as is 
physically evident for reflection from a semi-trans¬ 
parent layer of finite thickness. It is concluded, then, 
that for all practical purposes T may be substituted 


for the other wake indices, if the correction factor sec 
/3 is applied for oblique incidence of sound on semi¬ 
transparent wakes. 

Since the wake index 'k' applying to short-pulse 
echoes depends on the range and on the attenuation 
coefficient inside the wake, Table 2 gives a more de¬ 
tailed illustration of the influence of the variable 
parameters. The effect is seen to be quite small; there¬ 
fore it does not influence the interpretation of the 
experiments on the E. W. Scripps wake carried out 
on November 28, 1944, during which the two trans¬ 
ducers referred to in Table 3 were employed. 

As far as . he range is concerned, there is a distinc¬ 
tion between short and long pulses. In order to obtain 
short-pulse echoes of a kind that can be treated by a 
simple acoustic theory, the sound beam must be 
trained perpendicularly at the wake and the range 
must be shorter than 200 times the signal length; 
equation (35) of Section 33.1.2, which formulates 
this condition in an exact manner, shows that the 
exact factor is a function of the directivity pattern 
of the transducer. Short-pulse echoes produced with 
a sound beam trained obliquely at the wake defy 
any simple mathematical analysis and, therefore, are 
of little use in the study of wakes. As regards long- 
pulse echoes, however, the range is of minor im¬ 
portance, and the aspect of the wake is of no conse¬ 
quence whatever because the dependence of the wake 
index on the aspect angle (3 is fully taken into account 
by equations (40) and (41). In practice, it should 
suffice to keep the ratio of wake width iv to range r 
less than about 0.1; this value of w/r makes it possible 
to neglect the inverse-cube correction factor appear¬ 
ing in equation (15), which was omitted from there on. 

33.2.2 Pulse Length 

Pulses varying in length from 0.3 to several hun¬ 
dred milliseconds have been employed in echo rang¬ 
ing at wakes. There are some general considerations 
concerning signal length that apply primarily to the 
tactical use of wake echoes. In practice, the design 
of the keying circuits and the build-up time of the 
transducer set a lower limit to the pulse length. While 
so far no special study has been made of the opt imum 
conditions for recognition of wake echoes, it may be 
surmised, from experience with echoes from finite 
targets, 3 that signals shorter than 10 msec are not 
suitable for satisfactory recognition by ear. But wake 
echoes obtained with 1-msec signals, and even with 
shorter ones, are readily recognized on sound range 









ECHOES FROM SUBMARINE WAKES 


523 


recorder traces and oscillograms. Reverberation, par¬ 
ticularly at long ranges, imposes an upper limit to the 
practicable pulse length. 

Rather different considerations govern the choice 
of pulse length for fundamental research into the 
physical constitution of wakes. The aim of such work 


which the internal density distribution is undergoing 
all the time, aside from echo fluctuations due to ran¬ 
dom interference and variable transmission loss. Only 
by averaging numerous instantaneous profiles could a 
truly representative picture of the n(x) distribution 
be obtained. 


Table 3. Wake indices. 


Transducer 

JK 

GD 1143 

2 <j e nr 

24 kc 

40 kc 

50 kc 

60 kc 

* 

8 

II 

© 

-7.48 db 

— 6.35 db 

-7.36 db 

-7.39 db 


-7.41 

-6.22 

-7.28 

-7.30 

5 


-7.36 

-6.14 

-7.22 

-7.25 

10 


-7.31 

—6.05 

-7.17 

-7.19 

15 


-7.26 

-5.96 

-7.12 

-7.13 

20 

(0 = 0°) 

-7.20 

-5.87 

-7.06 

-7.08 

25 


-7.15 

-5.77 

-7.00 

-7.02 

30 


-7.03 

— 5.57 

-6.88 

-6.89 

40 


-6.90 

-5.35 

-6.75 

-6.76 

50 


may be either to establish the overall properties of a 
wake, or to resolve its microstructure. In the first 
case the use of long signals, overlapping the entire 
wake, is indicated, whereas in the second case maxi¬ 
mum resolving power is achieved by extremely short 
pulses. According to equation (40) the wake strength 
determined with long pulses is a function of (1) the 
depth of the wake, (2) the average cross section for 
scattering and extinction by the bubble population, 
and (3) the acoustic thickness a e X(w) of the entire 
wake, which may be determined quite independently 
by measurement of the horizontal transmission loss. 
Therefore, a simultaneous observation of the echoes 
returned by the wake and of the horizontal transmis¬ 
sion loss through it offers the greatest promise for 
testing the adequacy of equation (40) for long signals. 
The corresponding equation (39) for short pvdses has 
been derived by neglecting the microstructure of the 
wake, by putting n(x) = constant — n. However, on 
inspection of the rigorous equation (24), it will be 
seen that the echo profile on the oscillogram is es¬ 
sentially proportional to the function 

n{x)e- 2a ’" (x) 

for the case of extremely short signals, and of an ideal 
sharp sound beam which could be realized approxi¬ 
mately by placing the transducer very close to the 
wake. Such an analysis of the microstructure of 
wakes by short-pulse echoes would be of rather 
limited practical value, because of the rapid changes 


As to signals of intermediate length, it may be pre¬ 
sumed that equation (39) will represent the variation 
of W with r 0 reasonably well. 

33.2.3 Frequency 

Most echo ranging at wakes has been carried out 
with frequencies between 20 and 60 kc. The available 
observations suggest a conspicuous variation of wake 
strength with frequency, but no such dependence can 
be anticipated theoretically. The dominant factor 
cTs/<ie in the formula for the wake strength does not 
change much with frequency, for bubbles of resonant 
size. At present little is known about the relative 
proportion of resonant bubbles in the total popula¬ 
tion and how this proportion changes with time; but 
there is no definite reason to believe that bubbles of 
nonresonant size predominate, in which case <r s /<r e 
would vary more markedly with frequency. In any 
event, the influence of cr 8 /<r e on the wake strength 
would not be expected to account for more than a few 
decibels. However, some frequency effect may result 
from the factor (1 — e _2<r ' A( “’ ) ), provided that the 
wake is highly transparent; otherwise the exponential 
would be small compared with I. 

33.3 ECHOES FROM SUBMARINE WAKES 

Quantitative data on the strength of submarine 
wakes have recently been computed from the original 
measurements of echo levels. Some of these have been 















524 


OBSERVATIONS OF WAKE ECHOES 


published before ; 4,5 others were obtained from the 
files of the San Diego laboratory. During these ex¬ 
periments, the echo-ranging vessel overtook the sub¬ 
marine while proceeding on a parallel course; the 
observations comprise surface runs, dives, and sub¬ 
merged level runs. In order to reduce the uncertain- 


ular Q3 = 0 degrees) and oblique (/? = bO degrees) 
incidence of the sound beam on the wake. With a few 
exceptions illustrated in Figures 5 and 6, the obser¬ 
vations did not extend over sufficiently long periods 
of time to reveal the gradual decay of the wakes. 
Hence only average values of the wake strength W 


Table 4. Submarine wake strengths. 



Run 

(3 in 
degrees 

*0 

in db 

Ping 
length 
in msec 

Frequency 
in kc 

Wake strength W 
in db 9.5 knots 
surfaced 

Wake stren 
6 knots si 

Depth 45 ft 

gth W in db 
lbmerged 

Depth 90 ft 

Average 
distance 
astern 
in yd 

USS S-23 

1 

0 

-9.4 

30 

60 

-18 


-28 

400 

(SS128) 

2 

0 

-9.4 

30 

60 

-16 


-25 

400 


3 

0 

-9.4 

30 

60 

-19 


-26 

400 







Avg -18 


Avg —26 


USS S-34 

1 

0 

-8.6 

30 

45 

-12 



300 

(SS139) 

2 

uo 

-6.5 

30 

45 

-14 



300 







Avg -13 





1 

0 

-8.6 

30 

45 



-22 

200 


2 

60 

-6.5 

30 

45 



-24 

200 









Avg -23 


USS Tile fish 

1 

0 

-8.6 

30 

45 

-15 


-22 

600 

(SS307) 

2 

0 

-8.6 

30 

45 

-13 


-20 

600 


3 

60 

— 6.5 

30 

45 

-11 


-19 

600 







Avg —13 


Avg -20 


USS S-18 

1 

60 

-6.5 

10 

45 


-34.6 


200 to 250 

(SS123) 




30 

45 


-32.6 


200 to 250 





100 

45 


-30.6 


200 to 250 








Avg -32.6 




2* 

60 

-6.5 

10 

20 


-21.8 


10 to 500 





30 

20 


-18.7 


10 to 500 





100 

20 


-16.7 


10 to 500 








Avg —19.0 




2* 

60 

—6.5 

10 

20 


-23.6 


650 to 850 





30 

20 


-21.0 


650 to 850 





100 

20 


-19.3 


650 to 850 








Avg —21.3 




2* 

60 

-6.5 

10 

20 


-1.8 


Decay of wake 





30 

20 


-2.3 


Decay of wake 





100 

20 


-2.6 


Decay of wake 








Avg —2.2 




* This run ia illustrated in Figure 1. Absolute values of the wake strength \V are uncertain because of lack of adequate calibration. 


ties of the relative position during the submerged 
portions of the runs, the submarine towed a marker 
buoy. Pelorus bearings on this buoy were logged from 
the echo-ranging vessel. With the aid of the original 
logs, a diagram was constructed for each run, giving 
the relative position of submarine and measuring- 
vessel. The distances behind the submarine to which 
wake echoes belonged were then read from these dia¬ 
grams. Measurements were made both for perpendic- 


are given in Table 4, together with the approximate 
distances astern to which they refer. The transducers 
used had a narrow directivity pattern horizontally 
and a very wide pattern vertically, so that even dur¬ 
ing the deepest dives — to 400 feet — there was no 
significant loss of sensitivity. 

Although the 0 point of the W scale in Figure 5 is 
rather uncertain because an adequate calibration of 
the sound gear is lacking for that particular day, the 














































ECHOES FROM SUBMARINE WAKES 


525 



Figure 5. Dependence of wake strength on distance astern. Plot for USS S-18, submerged to a depth of 45 ft, for run 2 
of Table 1. Echo-ranging vessel and submarine were proceeding on parallel courses at constant speeds of 8 and 6 knots 
respectively. 


plot illustrates some significant features of the obser¬ 
vations. The individual points of the diagram are 
computed from the average of five successive echo 
levels, and the scattering of these averages gives a 
good idea of the magnitude of echo fluctuations en¬ 
countered in practice. Signals 10, 30, and 100 msec 
long were sent out in cyclic succession, so that the 
three curves for the different pulse lengths refer to the 
same wake. Despite the large echo fluctuations, there 
is good evidence for an increase of W with the signal 
length r 0 . The steep rise of the curves at zero distance 
astern probably is due to the stern of the submarine. 


Up to 500 yd astern — corresponding to a wake age 
of 2 minutes — the wake strength changes very little, 
if any. But when the observations were resumed at 
G70 yd astern, the decay of the wake had definitely 
set in. The values given in Table 4 suggest that for 
this wake the decay rate increased with increasing 
pulse length. 

All reliable numerical values of W have been col¬ 
lected in Table 4, together with the values of the wake 
index used in the individual computations. The latter 
will permit the computation from W of the corre¬ 
sponding target strength of the wake, if desired. The 







































526 


OBSERVATIONS OF WAKE ECHOES 


outstanding feature of the table is the greater strength 
of the wake laid by surfaced submarines, compared 
with those from submerged runs. However, during a 
dive the wake strength does not decline steadily. In¬ 
stead, repeated peaks occur. Some of the peak values 
even equal the strength of the surface wakes, as il¬ 
lustrated in Figure 6. These peaks are undoubtedly 
connected with the diving operations, movement of 
diving planes, blowing of tanks, and other operations. 
While the surface values of IF are surprisingly con¬ 
sistent — about — 15 db — only the order of magni¬ 
tude of the subsurface strength can be regarded as 
established, perhaps —25 db to —30 db. The relative 
acoustic weakness of wakes behind submerged sub¬ 
marines probably results from several causes, such as 
lack of entrained air and the reduction of cavitation 
and bubble production at the higher pressure. A small 
but definite increase of IF with pulse length as the pulse 
length changes from 10 to 100 msec is found for both 
submerged runs of the USS S-18 (SS123). The in¬ 
crease is small and results largely from the extension 
of the wake along the axis of the sound beam. Even 
for a wake whose thickness is less than the signal 
length, the echo will vary with signal length when the 
transducer is pointed obliquely at the wake. Only 
for normal incidence of the sound beam is the change 
of target strength with pulse length a simple, readily 
predictable effect. 

33.4 ECHOES FROM SURFACE 
VESSEL WAKES 

The San Diego group has studied echoes from 
wakes laid by numerous surface craft. Early experi¬ 
ments were carried out in San Diego harbor. During 
1944, the group carried out a large program of record¬ 
ing echoes from the wakes of a number of surface 
vessels, including aircraft carriers, destroyers, and 
some small craft. Wakes for this program were laid on 
the open sea off San Diego. 

33.4.1 Echo Ranging at Wakes in 
San Diego Harbor, 1943 

For these experiments an echo-ranging transducer 
was mounted on a barge moored to one side of the 
harbor channel. 1 Most of the measurements were 
made on wakes laid by a motor launch (length 40 ft, 
beam 11 ft, draft 2}+ ft) traveling at 4 to 6 knots. 
Incidental results were also obtained by echo ranging 
at wakes produced by other vessels which happened 


to pass; these vessels probably did not travel at full 
speed in the harbor. 

The chief interest of these experiments, which have 
been reported in detail in reference 1, lies in the fact 
that short signals — only 9 msec long — were trans¬ 
mitted alternately at 15, 24, and 30 kc. Thus it is 
possible to analyze the results for a possible depend¬ 
ence on frequency both of the wake strength and its 
decay rate. The absolute values of the wake strength 
appear to be less reliable, for two reasons. First, 
difficulties with the calibration of transducers seem 
to have been experienced during the early phases of 
the San Diego wake studies; such would affect the 
absolute values of IF, without impairing the results 
concerning the dependence of IF on frequency. Sec¬ 
ond, the measurements in the shallow waters of San 
Diego harbor are likely to have been disturbed by 
bottom-reflected sound; indeed, an apparent de¬ 
pendence of IF on the range was explainable only 
as caused by some peculiarity in the bottom contour. 

The results of these early measurements are sum¬ 
marized in Tables 5 and 6. Values of the wake st rength, 
obtained with 9-msec signals at three different fre¬ 
quencies, are collected in Table 5, which also contains 
the attenuation coefficients a and the wake indices 4> 0 
used in these computations. 

There is no information available on the wake age 
at which the observations on the larger vessels were 
made; probably the age did not exceed a few minutes, 
and the initial wake strength had decayed only 
slightly. The decay of the wakes laid by the launch 
was studied systematically over a period of 12 min¬ 
utes, after which time the echo intensity had dropped 
to the reverberation level. While the decay of the 
15-kc echoes appeared to start immediately after pas¬ 
sage of the launch, the echoes at 24 and 30 kc main¬ 
tained their initial strength for about 2 minutes before 
they began to decay. The decay of the echo intensity 
follows a simple exponential law, to a good approxi¬ 
mation; thus the strength of the echo expressed on a 
decibel scale decreases linearly with time. The decay 
rates found are listed in Table 6. Within the errors of 
observation, there is no dependence of the decay rate 
on frequency. But the wake strength IF seems to in¬ 
crease with frequency. From the average IF for each 
frequency of the vessels contained in Table 5, exclud¬ 
ing the launch and the three fishing boats, the follow¬ 
ing differences are found: 

II 24 — IF is = -(-1.0 db 

IF 30 — IF 24 — +4.6 db. 



ECHOES FROM SURFACE VESSEL WAKES 


527 


DIVE BEGINS WAKE AT 

VENTS OPEN 90 FT 



TIME IN SECONDS 

Figure 6. Wake strength and distance astern. 


Table 5. Surface vessel wake strengths. 




W in db for 9-msec pulses 

Vessel 

Range 

in 

yd 

15 kc 

a = 3.0 db per kyd 
= -5.8 db 

24 kc 

a = 5.0 db per kyd 
= -8.0 db' 

30 kc 

a = 7.0 db per kyd 
= -8.4 db 

40-ft motor launch 

60-150 

- 2.9 

+ 3.1 

+ 8.4 

Tanker 

330 

+ 8.1 

+ 7.7 

+ 9.0 

Fishing boat 

298 

+ 0.7 

+ 10.7 


Fishing boat 

150 

0.0 

- 0.6 


Fishing boat 

152 


- 0.4 


Kelp barge 

140 

+ 7.3 

+ 5.5 

+ 12.2 

Kelp barge 

190 

+ 4.5 

+ 9.1 

+ 14.3 

50-ft boat 

170 

+ 2.7 

+ 7.9 

+ 12.9 

Transport 

450 

+ 18.9 

+ 17.5 

+22.9 

Tank boat 

580 

+ 10.9 

+ 10.3 

+ 14.7 

Avg (excluding launch 
and fishing boats) 


+ 8.7 

+ 9.7 

+ 14.3 


The same trend is definitely established by the values 
of W for the 40-ft launch in Table 5, which are the 
averages resulting from 16 wakes. 

33 . 4.2 Deep Water 

By 1944 considerable progress had been made in 
standardizing the sound gear at frequencies in the 


neighborhood of 24 kc. It is believed that the relia¬ 
bility of the absolute calibration of the transducers 
used for these later measurements is much greater at 
these frequencies than during the earlier measure¬ 
ments. Moreover, this program was executed in deep 
water off San Diego, so that interference from bottom- 
reflected sound was avoided. Pending a comprehen- 



















































528 


OBSERVATIONS OF WAKE ECHOES 


sive report on these investigations, a summary of pre¬ 
liminary results has been made available for the 
purposes of this volume. 

The experiments with craft other than the USS 
Jasper (PYcl3) itself followed a single pattern. The 
recording vessel, the Jasper , was lying to in the open 
sea, and the wake vessel approached and passed her 
while maintaining a straight course. As the wake 
vessel approached, the Jasper echo-ranged on her, 
training the sonar projector with the aid of a pelorus 
manned on the flying bridge. When the wake vessel 
came abreast of the Jasper at the time of closest ap¬ 
proach, the training of the sonar projector was halted, 
and its true bearing was held fixed and approximately 


Table 6. Decay rate of wake of 40-ft launch. 


Frequency 
in kc 

Mean decay rate 
and its prob¬ 
able error in 
db per minute 

Standard 
deviation in 
db per minute 

Number 

of 

wakes 

15 

6.8 ± 0.6 

4.0 

20 

24 

7.0 ± 0.4 

2.5 

20 

30 

7.0 ± 0.5 

3.0 

20 


perpendicular to the wake until the end of the run. 
When possible, recording was continued, either con¬ 
tinuously or intermittently, until no more echoes 
from the wake were detectable above the background 
of reverberation. 

When the Jasper was studying her own wake, a 
different technique was necessarily adopted. In this 
case, the Jasper ran on a straight course at 12 knots 
for 10 or 15 minutes; then she turned around and ran 
back parallel to her original course with her sonar 
projector trained abeam so that the sound beam was 
directed normal to the original course. Recording was 
continued until a wake echo was no longer discernible 
above the reverberation. 

A sample graph of the peak echo level received 
from a wake against time is shown in Figure 7, which 
also contains the echo levels received from a sphere 
3 ft in diameter buoyed behind the wake at a depth 
of 6 ft. On this run three different signal lengths, re¬ 
peated in cyclical succession, were used to give wake 
echoes. This system of interchanging signal lengths 
was used on a number of wakes. On some occasions 
the gear used for cycling the pulses was not in order, 
and five or six echoes were recorded at one pulse 
length before the pulse length was changed manually. 
On a few occasions only one pulse length was used 
throughout the run. 


The initial wake strength is determined by the 
initial echo level - the level of the wake echo at the 
time (zero time) when the stern of the wake vessel 
has just passed out of the sound beam. This time can 
only be estimated, and sometimes echoes were not 
recorded until some time after the wake was laid. 
In such cases the values to be used for initial echo 
level are obtained by extrapolating the observations 
available back to the zero time. The decay of the 
wake is measured as the slope in decibels per minute 
of the echo level-time curve. Some thought was given 
to the possibility of two decat’' rates in the wake, the 
dividing line between them being rather sharp in time, 
but the data were not sufficiently well defined to 
allow such a distinction to be made. Therefore, only 
one decay rate was obtained for each wake. This is 
the rate at which the echoes seemed to decay steadily 
for several minutes before they became indistin¬ 
guishable. 


m 

o 


<D 

O 


20 

0 


•20 

0 


-20 

-4 0 

0 2 4 6 8 10 12 

TIME IN MIN AFTER CROSSING SOUND BEAM 



Figure 7. Decay of wake from E. W. Scripps. Run 1 
of Table 8, 24 kc.' 


It seems fairly certain that there is a systematic 
increase in W with the size of the wake vessel, but 
Table 7 shows that the magnitude of the effect is not 
very large. All that can be said about the speed of 
the wake vessels during these experiments is that 
they seemed to be running within their normal range 
of operating speeds. 

These averages include echoes obtained both with 
10-msec and 30-msec pulses, as the number of avail¬ 
able data was small and the increase of W 30 msec over 
IF io msec is moderate (see Table 8). Presumably, the 
standard deviation would not be reduced much by 
separating the results according to signal length. 

The wake strength appears to increase with fre- 




































ECHOES FROM SURFACE VESSEL WAKES 


529 


Table 7. Dependence of wake strength on wake-laying vessel. 


Type of wake vessel 

24 kc 

60 kc 

Average W 
in db 

Standard deviation 
of IF in db 

Number of 
wakes 

Average W 
in db 

Standard deviation 
of W in db 

Number of 
wakes 

CVE’s and AP’s 

- 7.7 

4.1 

5 




DD’s and DE’s 

- 9.6 

6.3 

5 

+7.9 

1.1 

2 

Laboratory yachts 

-13.6 

2.6 

5 

+ 1.6 

3.0 

8 

(Scripps & Jasper) 







Small boats 

-18.2 

2.0 

2 

-3.7 

2.1 

2 


Table 8. Dependence of wake strength on pulse length. 


Type of 
wake vessel 

Frequency 
in kc 

Average 

difference of wake strength in db 

Total 
number of 
wakes 

W 10 msec — W 1 msec 

W 30 msec — WlO msec 

DD’s 

24 

6.5 

3.5 

3 

CVE’s 

24 

8.5 

3.0 

3 

E. W. Scripps 

24 

9.0 

1.5 

1 

E. W. Scripps 

60 

8.0 

4.0 

2 

USS Jasper 

24 

9.5 

3.0 

3 

(PYcl3) 





USS Jasper 

60 

7.5 

3.5 

3 

(PYcl3) 





Avg of all vessels 





for all frequencies 


8.2 

3.7 



quency. From Table 7, it can be seen that the average 
difference between wake strength at (JO kc and wake 
strength at 24 kc is 16 db. But the reality of this 
phenomenon is doubtful, because use of this same 
underwater sound equipment in measurements of tar¬ 
get strengths of submarines has yielded results at 60 
kc which are also 10 to 20 db above the 24 kc results, 
thus contradicting theoretical expectations (see Sec¬ 
tion 23.6.2). It is also important that measurements 
on submarine wakes made with different equipment 
and discussed before (see Table 4) show a decrease of 
W with increasing frequency rather than an increase. 
In a separate set of careful experiments on surface- 
vessel wakes, which are summarized in Table 9, a 
single instance was found where there was a marked 
difference of opposite sign between wake strength at 
00 kc and wake strength at 24 kc. The existence of an 
isolated but well documented instance like this where 
an apparent trend is contradicted must be given con¬ 
siderable weight when conclusions are drawn about 
the frequency dependence of the wake effect. 

The decay rate of surface wakes shows very little 
dependence on frequency between 24 and 60 kc. The 
average decay rates of the wakes described above are 
1.3(5 db per minute at 24 kc, and 1.18 db per minute 



AGE OF WAKE IN MINUTES 


Figure 8. Echo level as function of age of wake, for 
various ping lengths at 24 kc. Wake vessel: Small 
carrier at about 15 knots. 

at 60 kc. The standard deviations of these measure¬ 
ments are 0.59 db per minute at 24 kc and 0.69 db per 
minute at 60 kc. This difference in averages is so much 























































530 


OBSERVATIONS OF WAKE ECHOES 


smaller than the spread of the data that it cannot be 
said that there is any significant difference between 
the decay rate at GO kc and the decay rate at 24 kc. 

Figure 8 shows a typical example of the variation 
of the echo level, or of the wake strength, with signal 
length. Numerical values of the variation of the echo 
level, or of the wake strength, with signal length are 
summarized in Table 8. 

This dependence of W on the signal length was pre¬ 
dicted from general theoretical considerations (see 
Section 33.1.2), and the observed magnitude of the 
effect permits an estimate of the average concentra¬ 
tion of bubbles in a wake (see Chapter 34). The num¬ 
ber of observed data is too small to warrant any con¬ 
clusions as to the influence of frequency and size of 
the wake vessel on the pulse length effect. 


Table 9. Wake strength and decay rate, E. W. Scrip pH. 


Run 

F requency 
in kc 

Wake 
index 
in db 

Wake strength 
W in db at age 
0-2 minutes 
3-msec pings 

Decay rate of 
wake strength 
dW/dt 

in db per minute 

1 

24 

-7.0 

— 5 

-1.5 

2 

24 

-7.0 

-it 


3 

60 

-7.0 

-21 

-0.7 

4 

60 

-7.0 

-16 


4 

45 

—6.5 

-22 

-0.7 


Table 9 contains some additional values of W for 
several wakes laid by E. IF. Scripps on a day when 
the sea was unusually calm. Experimental details 
concerning these observations have already been 
given in Section 32.3.2, and the transducers used are 
listed in Table 3; the echo level-time curve of Run 1 
of Table 9 is reproduced in Figure 7. 

The average W at 24 kc (mean of Runs 1 and 2) is 
— 8 db for 3-msec pulses. In order to make this value 
comparable with the average value of W at 24 kc for 
the wakes of the Scripps and Jasper in Table 7, which 
is —13.6 db, a correction for the difference in signal 
lengths used must be made; from Table 8 it may be 
estimated that IFiomsec— IFimsec is of the order of 
+6 db. The corrected TF 24 kc of Table 9 is then — 2 db, 
or about 12 db greater than TF 24 kc in Table 7. After 
the corresponding correction of the GO-kc data of Run 
3 has been made, TFeo kc in Table 9 is still 17 db 
smaller than its counterpart in Table 7; the origin of 
this serious discrepancy remains unexplained. As for 
the minor discrepancy at 24 kc it seems worth men¬ 
tioning that the E. W. Scripps had been outfitted 
with a new propeller and engine in the fall of 1944, 


so that the data in Tables 7 and 9 are not strictly 
comparable. The decay rates in Table 9 do not differ 
significantly from the averages for all surface vessels 
quoted before. 

33.5 ECHOES FROM MODEL PROPELLER 
WAKES 

At the Woods Hole Oceanographic Institution, 6 a 
number of experiments were made on the scattering 
of sound by the wakes of stationary model propellers. 
Although the published data do not yield absolute 
values of the wake strength, they give some interest¬ 
ing information on the relative echo intensity as a 
function of the frequency of sound and of the depth 
of the propeller. 

In order to measure the scattering, the hydrophone 
and transducer were mounted on the same side of the 
wake in a horizontal plane including the wake axis. 
The axis of the hydrophone was vertical and the 
transducer was directed toward the wake. Both 
instruments were secured to a pipe frame and were 
separated by a baffle, in order to reduce the passage 
of the direct signal from the transducer to the hydro¬ 
phone. The baffle consisted of a sheet of Celotex 32 in. 
square and x /2 in. thick, sheathed with copper; the 
plane of this sheet was perpendicular to the axis of 
the wake. This single baffle was found to be preferable 
to a wedge-shaped baffle composed of two sheets of 
Celotex making an angle with one another. In order 
to reduce the direct signal still further, the hydro¬ 
phone was partially enclosed in a box lined with 
Celotex and open on the side toward the wake. The 
perpendicular distance from the instruments to the 
wake axis was 5 ft; the plane of the baffle, midway 
between the instruments, was 10 ft from the plane 
parallel to it through the propeller. With this arrange¬ 
ment, scattering measurements were made in the 
deep spot 200 ft off the wharf at depths varying from 
5 to GO ft and at frequencies from 30 to GO kc. At 
lower frequencies the reflection was too small to 
measure. 

Each determination of the scattering involved the 
measurement of the signal at the hydrophone under 
three conditions: (1) with the propeller at rest and 
the transducer on; (2) with the propeller running and 
the transducer on; (3) with the propeller running and 
the transducer off. The results of these three measure¬ 
ments, in decibels, will be referred to by Zi , z <>, and z 3 , 
respectively, with Zi representing the direct signal 
from the transducer in the absence of scattering — 












ECHOES FROM MODEL PROPELLER WAKES 


531 



Figure 9. Dependence of sound scattered from 10- 
inch propeller at 1,600 rpm on depth below surface. 
Direct signal constant for each frequency. 


the sound which travels around and through the 
baffle, and z 3 representing the cavitation and pro¬ 
peller noise. The true value of the scattered sound in 
decibels, which we call z r , is in general different from 
z 2 but may be obtained from it by correction for the 
direct signal (zO and for cavitation and propeller 
noise (z 3 ). It is given by the equation 

10 Zr/l ° = 10” Ao - 10 2l/l ° - 10* ,/l0 . 

The results presented below were calculated in this 
way. It should be pointed out that the effect of z 3 was 
in all cases negligible. 

An interval of a minute or a minute and a half was 
always allowed between successive determinations to 
make sure that there should be no residual wake from 
the previous determination to interfere with the fol¬ 
lowing one. Only the 10-in. and 14-in. propellers were 
used, and only at the highest speed, 1,000 rpm. 
Under other conditions the scattered sound was too 
small to measure satisfactorily. 

The results of the study are shown in Figures 9 and 
10. Although the scatter of the observations is large, 
particularly with the 14-in. propeller, there can be no 
question of the general effect. It is evident that there 



Figure 10. Dependence of sound scattered from 14- 
inch propeller at 1,600 rpm on depth below surface. 
Direct signal constant for each frequency. 


is a marked decrease in sound scattering with depth. 
At a frequency of 00 kc the scattered sound is less 
than 1 fo as much at 60 ft as at 5 ft. In this respect 
the situation is similar to that observed in the case of 
attenuation (see Section 32.5). 

The data plotted in Figures 9 and 10 give simply 
the total reflected sound in decibels at the hydro¬ 
phone. They take no account of the strength of the 
direct signal from the transducer. Since the oscillator 
was always set to give the same output, this signal 
may be regarded as constant for each frequency. 
Consequently at each frequency the change in the 
decibel level of the reflected signal with depth gives 
the change in the scattering coefficient. Nevertheless, 
in order to obtain absolute values of the scattering 
coefficient and to discover its dependence on fre¬ 
quency it is necessary to take into account the 
strength of the direct signal which would be received 
by the hydrophone in the absence of a wake at the 
position of what may be called the “virtual image” of 
the hydrophone with respect to the wake. This is a 
point at the same distance from the wake as the 
hydrophone, but on the opposite side of it. It was 
estimated to be 0 ft away from the transducer. With 
this in mind, throughout the study, daily determina¬ 
tions were made of the response of the hydrophone 



















































532 


OBSERV VTIONS OF W AKE ECHOES 


6 ft in front of the transducer and in the same orienta¬ 
tion as in the actual measurements. Such determina¬ 
tions were made for each frequency used in the meas¬ 
urements. The results were found to be independent 


Table 10. Direct signal as function of frequency. 


Frequency in kc 

30 

40 

50 

60 

Direct signal z o 

22.5 

26.2 

34.0 

36.8 


of depth, as would be expected, and were reasonably 
constant from day to day. Relative minor variations 
are probably attributable to small differences in the 
spacing of the two instruments. Values of the direct 
signal, which will be called z 0 , measured in this way 
are given in Table 10. On the basis of these results, 
it is a simple task to calculate the relative intensity 
of the reflected sound. This, in terms of decibels, is 
simply z r — z 0 . Table 11 gives the values of z r — z a 
obtained with each of the two propellers at a depth 
of 10 ft. In arriving at these results values of z r were 
read off the smooth curves of Figures 9 and 10; 
values of z 0 were taken from Table 10. 


The intensities of the scattered sound at other 
depths are, of course, less than these, in accordance 
with the way in which the curves of Figures 9 and 10 
drop off. It is evident that there is no considerable 


Table 11. Reflected sound as function of frequency. 



10-in. 

propeller at 1,600 rpra and 
depth of 10 ft 

Frequency in kc 

30 

40 

50 

60 

z, — Zo 

-24.5 

-25.2 

-24.0 

-20.8 



14-in. 

propeller at 1,600 rpm and 
depth of 10 ft 

Frequency in kc 

30 

40 

50 

60 

Z r — Zn 

-22.0 

-22.2 

-24.0 

-22.8 


effect of frequency between 30 kc and 00 kc. The de¬ 
crease of the echo intensity with depth is again a 
manifestation of the influence of increased pressure 
on the formation and dissolution of bubbles, as in the 
decrease of the attenuation with depth described in 
Section 32.5. 































Chapter 34 


HOLE OF BUBBLES IN ACOUSTIC WAKES 


T he previous chapters have developed a general 
theoretical background for the study of wakes 
and have presented the results of acoustic measure¬ 
ments on wakes. In this chapter, a review is first 
given of the evidence that bubbles are the chief source 
of the acoustic properties of wakes. Next, the quanti¬ 
tative acoustic measurements are compared with the 
theoretical formulas derived in Chapter 28. From this 
comparison, conclusions are drawn as to the amount 
of air present in wakes. Finally, the rate of decay of 
acoustic wakes is discussed, and shown to be roughly 
similar to the rate at which air bubbles disappear in 
sea water. 

34.1 EVIDENCE FOR AIR RUBBLES IN 
WAKES 

At the present time it seems almost certain that 
small air bubbles are responsible for the observed re¬ 
flection and absorption of sound by surface ship and 
submarine wakes. The evidence for this is of two 
general types, qualitative and quantitative. 

From a qualitative standpoint, air bubbles provide 
the only mechanism yet proposed which can explain 
the general behavior of wake echoes. In particular, 
no other explanation seems capable of explaining the 
very marked dependence of scattering and absorbing- 
power on the depth of the wake. The measurements 
with the model propeller, described in Sections 32.5 
and 33.5, show unmistakably a pronounced weaken¬ 
ing of both attenuation and scattering when the pro¬ 
peller is below about 30 ft. Measurements of echoes 
from submarine wakes show a similar decrease of 
about 5 to 10 db in wake strength when the sub¬ 
marine dives from the surface to periscope depth. 
Practical echo-ranging trials confirm the disappear¬ 
ance of wake echoes when the submarine dives below 
200 or 300 ft. These observations cannot be explained 
on the assumption that turbulence or temperature 
effects are responsible for the acoustic properties of 
wakes, but they follow naturally from the assumption 
that bubbles are the important agents. 


From a quantitative standpoint, the magnitude of 
the observed effects is enormously greater than can 
apparently be explained by any assumed mechanism 
besides the presence of small bubbles in the wake. It 
has already been noted, in Chapter 29, that on the 
basis of present acoustic theory, neither turbulence 
nor temperature irregularities could account for any 
appreciable scattering or attenuation by wakes. The 
absorbing and scattering power of a single resonant 
bubble, analyzed in Section 28.1, is so great, how¬ 
ever, that a relatively small number of bubbles is 
required to explain the observed acoustic effects. 

Any theory of the acoustic properties of wakes can¬ 
not be regarded as completely confirmed until reliable 
quantitative data are shown to be in close numerical 
agreement with the theoretical predictions. Until in¬ 
dependent nonacoustic measurements are made of 
the bubble density in wakes, or until accurate and 
reproducible acoustic data can be obtained on wakes 
under a variety of conditions, it is not possible to 
verify the “bubble hypothesis” explaining the origin 
of the acoustic wake. Nevertheless, the general evi¬ 
dence seems sufficiently strong to make this hy¬ 
pothesis highly probable. 

34.2 TRANSMISSION THROUGH WAKES 

The attenuation of sound by air bubbles in water 
has been discussed in Section 28.2. The conclusion 
reached was that probably most of the attenuation 
is produced by bubbles whose radii are close to the 
radius R r of a resonant bubble. Integrating the con¬ 
tributions to the attenuation from all bubbles near 
resonant size leads to equation (67) of Chapter 28 for 
K e , the attenuation coefficient in decibels per yard: 

K e = 1.4 X 10 5 a(tf r ) , (1) 

where u(R r )dR is the volume of air per cu cm in 
bubbles whose radii lie between R r and R r + dR. 
If K e is known at all frequencies, equation (1) gives 
u(R r ) for bubbles of any radius. The total volume u 


533 


534 


ROLE OF BUBBLES IN ACOUSTIC WAKES 


of air in one cu cm of water is then given by the 
integral 

ftmax 

u = J u(R r )dR r , (2) 

o 

where /f max , the maximum radius of any bubble 
present, is assumed to be much less than 1 cm. 

Since the attenuation coefficient K e is directly pro¬ 
portional to the bubble density u(R r ), and since also 
the damping constant 5 discussed in Section 28.2 does 
not affect K e , measurements of acoustic attenuation 
provide a sensitive determination of the amount of 
air present in wakes. The actual attenuations ob¬ 
served, however, are somewhat complicated by the 
geometry, since the wake is never sufficiently deep to 
ensure that no sound reaches the measuring hydro- 


To find the absorption in decibels per yard, the re¬ 
sulting transmission losses have been divided by the 
wake widths for the destroyers given in Section 
31.3.1. The values of K e for a destroyer speed of 15 
knots are listed in Table 1, together with the cor¬ 
responding values of u(R r ). The values of u(R r ) for 
different ages of the wakes were plotted against R r 
for destroyer speeds of 10, 15, 20, and 25 knots, re¬ 
spectively, and the areas under these curves were 
determined by graphical integration. The resulting 
values of u, the relative amount of air present, in 
bubbles of all sizes for different destroyer speeds and 
wake ages are given in Table 2. Starred values are 
uncertain, since they are based primarily on the 
extrapolated parts of the graphs. 

Apparently no direct estimates have been made of 
air present as bubbles in destroyer wakes. The only 


Table 1. Attenuation coefficient and density of resonant bubbles—destroyer at 15 knots. 



Age of wake and distance astern 



1 

minute 

3 minutes 


5 minutes 


Frequency 

500 yd astern 

1,500 yd astern 

2,500 yd astern 

Rr 

in kc 

K e 

u(Rr) 

K e 

v(R r ) 

Ke 

u(R r ) 

in cm 

3 

0.35 

2.5 X 10-6 

0.03 

2.1 X 10~ 7 



0.107 

8 

0.67 

4.8 X 10- 6 

0.21 

1.5 X 10-8 

0.03 

2.1 X 10“ 7 

0.040 

20 

1.11 

7.9 X 10-s 

0.48 

3.4 X 10-8 

0.22 

1.6 X 10-8 

0.016 

40 

1.65 

1.18 X 10“ s 

0.79 

5.6 X IQ" 6 

0.43 

3.1 X 10-« 

o.oos 


phone below the wake. As a result of this uncertainty, 
the bubble densities found by use of equations (1) 
and (2) are somewhat indefinite, though they are 
probably not in error by a factor of more than two. 

Bubble densities may be computed from acoustic 
measurements for destroyers at different speeds and 
for different wake ages. They may also be computed 
for a small high-speed propeller with no forward 
motion. 

34.2.1 Wakes of Destroyers and 
Destroyer Escorts 

The computations for destroyers and similar ves¬ 
sels are based on the extensive transmission measure¬ 
ments across wakes reported in Section 32.3.1. The 
smoothed curves represented by equations (10) and 
(11) of Chapter 32 have been used, and an average 
taken for source outside the wake and source inside 
the wake, since these represent lower and upper 
limits to the absorption in the top 10 ft of the wake. 


Table 2. Fraction of air present as bubbles in de¬ 
stroyer wakes. 


Destroyer speed 
in knots 

u 

1 minute 

Age of wake 

3 minutes 

5 minutes 

10 

5.2 X 10- 7 * 

1.4 X 10- 7 

6.5 X 10- 8 

15 

7.4 X 10“ 7 * 

2.0 X 10- 7 

6.9 X lO- 8 

20 

7.0 X 10- 7 * 

2.3 X 10“ 7 

8.5 X lO” 8 

25 

9.1 X 10- 7 * 

2.1 X 10- 7 

8.7 X lO” 8 


* Uncertain. 


attempts to collect bubbles in ship wakes are ap¬ 
parently the attempts made with a 78-ft yacht. 1 
About 1 cu cm per minute of air was collected through 
a ring 8 in. in diameter, 6 ft behind a propeller 38 in. 
in diameter rotating at tip speeds between 50 and 60 
ft per second. Cavitation bubbles could be seen in the 
water, but the bubble density computed for a slip¬ 
stream speed of 5 ft per second is only 5 X 10~ 7 parts 
of air by volume in 1 part water. This value is in 

























ECHOES FROM WAKES 


535 


moderately good agreement with the values shown 
in Table 2 . 

3t.2.2 \\ akes of Model Propellers 

A similar computation may be carried out for the 
wakes of small propellers. Measured values of the 
absorption across a wake are reported in Section 32.5. 
The cross section of the wake was about 1.5 yd wide 
at the point where the measurements were made. The 
values of u(R r ) were computed by use of equation ( 1 ) 
for the 10 -in. propeller at 1,000 rpm and for depths 
of 10 ft, 20 ft, and 30 ft. Somewhat smaller values 
are found for the 14-in. propeller at the same rpm, 
possibly as a result of the narrower blades and lower 
pitch of this propeller. The corresponding values of 
u — the relative amount of air present in bubbles of 
all sizes, found directly from these curves —- are 
given in Table 3. 


Table 3. Fraction of air present as bubbles in wake 
of 10-in. model propeller. 


Depth in feet 

u 

10 

3 X 10-« 

20 

2 X 10-* 

30 

9 X 10- 7 


It is perhaps unexpected that the bubble density in 
the wake of a 10 -in. propeller be from five to ten times 
as great as the corresponding density in a destroyer 
wake. Further analysis shows this is not too surpris¬ 
ing. The propeller developed 11 hp during operation, 
with a tip speed of 70 ft per second. When a destroyer 
is making 15 knots, its two propellers with diameters 
between 9 and 11 ft, have a comparable tip speed, 
about 80 ft per second. Moreover the destroyer is 
moving rapidly, and it is well known that a propeller 
which is held stationary in the water tends to produce 
stronger tip vortices than one at the same rpm which 
pushes itself through the water. Thus the small pro¬ 
peller may be expected to cavitate more vigorously 
than the propeller of a destroyer at 15 knots. The 
volumes over which the bubbles produced in one 
second are spread in these two cases are proportional 
to the total propeller areas. Thus it would not be 
surprising to find that the bubble density measured 
behind the small propeller is greater than the cor¬ 
responding density in the destroyer wake. 

34.3 ECHOES FROM WAKES 

The wake strength W is related to the bubble den¬ 
sity in a more complicated way than is the attenua¬ 


tion coefficient K e . In addition, W depends both on 
the detailed geometrical properties of the wake, and 
on the physical properties of bubbles of different 
sizes, and cannot therefore be predicted with any 
exactness for a known distribution of bubbles. Thus, 
at most, a rather general agreement can be expected 
between observed and predicted wake strengths. 

The formulas are simplest for long pulses; when 
bubbles of a single size are present, the wake strength 
W for long pulses is given by the equation 

W = 10 log j— [1 - e - 2ffejV(u ’ ) ]j. ( 3 ) 

18 xcr e ) 

taken from equation (48) of Chapter 33. The quanti¬ 
ties (t s and <r«. are the scattering and absorption cross 
section defined by equations (34) and (43) in Chap¬ 
ter 28, while h is the depth of the wake, measured in 
yards. N(w) is the total number of bubbles in a col¬ 
umn one sq cm in cross section extending through the 
wake in a direction parallel to the sound beam [see 
equation (54) of Chapter 28], and the product 
a e N(io) is 0.23 times //,, , which is the total transmis¬ 
sion loss across the wake measured in decibels. Thus 
when this transmission loss is high, the exponential 
term is very small, and W approaches the limiting- 
value 

W = 10 log h + 10 log ( —— )• (d) 

\ 8 TT(T e / 

Equation (46) of Chapter 3 gives the ratio of cr s to 
a e in terms of <5, the so-called damping constant, and 
r), the ratio of bubble circumference to the wave 
length of the sound which represents the contribution 
of radiation damping to the damping constant. Values 


Table 4. Observed frequency dependence of ratio of 
scattering to extinction cross sections. 


Frequency in kc 

1 5 8 13 

19 

26 

36 

45 

10 lo s fe) 

-21 -22 -23 -24 

-25 

-26 

-27 

-28 


of these two quantities have been taken from Figure 2 
and equation (23) of Chapter 28 and the resulting 
values of 10 log (o-JSiroe) shown in Figure 1 and 
Table 4 of this chapter. At 24 kc, this quantity is —26 
db, and the maximum value of W is equal to 

W =10 log h — 26. (5) 

For a typical wake 10 yd deep this gives a maximum 
wake strength of —16 db. 











536 


ROLE OF BUBBLES IN ACOUSTIC WAKES 



0 10 20 30 40 90 60 70 


FREQUENCY IN KC 

Figure 1. Frequency dependence of ratio of scattering 
to extinction cross sections. 

Considering the systematic difference between the 
observed and theoretical values of 5, as evident in 
Figure 2 of Chapter 28, it appears highly probable 
that at 60 kc the damping constant will not be smaller 
than its theoretically predicted value, since the actual 
damping by dissipative effects should not be less than 


<x s /a e . However, the scattering and absorption cross 
sections of a resonant bubble are so much greater 
than those of other sizes that it seems unlikely that 
bubbles other than those near resonance can contrib¬ 
ute appreciably to either the scattering or the ab¬ 
sorption. Thus equation (5) may be used for actual 
wakes, provided that a value of appropriate to a 
resonant bubble is taken. 

On the other hand, when both the product a e N(;w) 
and the transmission across the wake are negligible, 
equation (3) gives for bubbles all of the same size the 
equation 

W = 10 log ftg)<r^ =1() log ig 6> (6) 

where w is the width of the wake in yards and n is the 
average number of bubbles per cubic centimeter. 
Since N (w) is the number of bubbles per square centi¬ 
meter appearing in projection on a plane perpendicu¬ 
lar to the sound beam, the equivalent product nw in 
equation (6) must have the same units — that is, 
square centimeters. It is customary to measure the 
wake width w in yards, or units of 91.5 cm. Hence, in 
order to keep equation (6) dimensionally correct, a 


Table 5. Frequency dependence of damping constant. 


Frequency in kc 

5 

10 

15 

20 

25 

30 

35 

40 

45 

10 log (3/85) 

-6.4 

-5.2 

-4.2 

-3.4 

-2.7 

-2.1 

-1.6 

-1.2 

-0.8 


that resulting from the flow of heat in and out of the 
oscillating bubble. This predicted value, derived from 
the theory given in Section 29.2, happens to be about 
one-third of the observed value of 8 at 24 kc. Hence 
the true damping constant for 60-kc sound very 
likely is greater than one-third of the observed damp¬ 
ing constant at 24 kc. This surmise implies that the 
theoretically predicted maximum wake strength for 
60-kc sound should not exceed the observed value of 
W at 24 kc by more than 5 db — because q is inde¬ 
pendent of frequency in this range — unless the ef¬ 
fective value of the wake depth h is quite different at 
the two frequencies. 

For the general case of a bubble population com¬ 
prising all sizes from the largest to the smallest, the 
analysis is more complicated. If many bubbles of very 
large radii are present, they will scatter without much 
absorbing, and <j a /a e will be increased. On the other 
hand if many bubbles of very small radii are present, 
these will absorb without much scattering, decreasing 


term 10 log 91.5, which is equal to +19.6, has been 
added to the right-hand side of equation (6) since w is 
measured in yards. When bubbles of varying sizes 
near resonance are considered, equation (6) is modi¬ 
fied by the substitution of S s for n<r 8 ; S s is a weighted 
mean of <x s for bubbles near resonance, according to 
equation (77) of Chapter 28, and is equal to 


Equation (6) then may be written 
W = 10 log h + 10 log w + 10 log u(R r ) 

+ 10 log + 19.6, (8) 

where h and w are the depth and width of the wake, 
respectively, both measured in yards. Values of 
10 log 3/85 are shown in Table 5 for resonant bubbles 
at different frequencies. In principle, equation (8) can 
be used to determine u(R r ) from the observed value 























ECHOES FROM WAKES 


537 


of W for any wake across which the transmission loss 
is less than 1 db. In practice, if the wake strength is 
less than about -30 db, the echo is difficult to dis¬ 
tinguish from the background. Since the theoretical 
maximum value of W is only -16 db for a wake 
10 yd deep, there is a relatively narrow spread of 
values over which u(R r ) can be varied to give meas¬ 
urable variations in W. 

34.3.1 Surface Vessels 

For surface ships the transmission loss across the 
wake is usually large. Thus in theory all surface 
wakes should exhibit a wake strength W given by 
equation (5). All wake strengths should be nearly 
constant and equal to — 16 db, except for small varia¬ 
tions in 10 log h, presumably not exceeding 3 db at 
most. 

An examination of the surface vessel wake 
strengths tabulated in Table 5 of Chapter 33 shows 
that the wake strengths are highly variable. The 
variability of transmission loss, which could not 
readily be taken into account in the measurements, 
probably accounts at least in part for this failure of 
the wake strengths to remain at a constant level. 

Even the average observed values of W, however, 
cannot be compared directly with the theoretical 
predictions. In the first place, the measured wake 
strengths all refer to peak amplitudes. Extensive 
measurements of reverberation records 2 show that 
the average peak amplitude is about 7 db higher than 
the average amplitude; these measurements refer to a 
segment of reverberation three to six times as long as 
the signal length. Moreover, since the rms amplitude 
is about 1 db above the average amplitude, it follows 
that —6 db should be applied as a net correction. 
According to the observations, this correction does 
not change rapidly in magnitude when the length of 
the reverberation segment analyzed is changed. Since 
echoes from wakes are structurally similar to rever¬ 
beration, it is concluded that a correction of —6 db 
applied to the observed values of W listed in Chapter 
33 presumably will suffice to express them on the in¬ 
tensity scale envisaged in equations (3) to (8). In 
addition, if surface-reflected sound reaching the wake 
is of the same intensity as the direct sound, the 
actual transmission anomaly is 3 db less than as¬ 
sumed; another 6 db should then be subtracted from 
the wake strengths reported in Chapter 33 to give the 
correct values. 

If the correction for surface-reflected sound is neg¬ 


lected, values of the observed wake strengths on an 
intensity scale may be found by subtracting 6 db from 
the values of W listed in Table 7 of Chapter 33. The 
resulting values are shown in Table 6, together with 


Table 6. Observed and predicted wake strengths. 



Observed wake 
strengths at 

24 kc in db 

h 

in yds 

Maximum theo¬ 
retical wake 
strength at 

24 kc in db 

CVE’s and AP’s 

-14 

15 

-14 

DD’s and DE’s 

-16 

8 

-17 

E. W. Scripps 

-20 

4.4 

-20 

USS Jasper 

-20 

6.7 

-18 

(PYcl3) 

Small boats 

-24 

2(?) 

— 23( ?) 


the wake depths h taken from Chapter 31 and the 
theoretical limiting values of W found from equation 
(5). The close agreement between theory and obser¬ 
vation for the larger vessels suggests that no large 
correction is required for the presence of surface-re¬ 
flected sound. This same conclusion is supported by 
agreement between direct and indirect determina¬ 
tions of submarine target strength at beam aspect, 
reported in Sections 21.5.4 and 23.8.1 of this volume. 

There are a few cases of anomalously high wake 
strengths, discussed in Section 33.4. These are diffi¬ 
cult to explain on the basis of scattering by bubbles. 
One possible effect worth considering, that could in 
principle give rise to very high wake strengths, is the 
specular reflection of sound from wakes. As pointed 
out in Section 28.3.4, bubbles not only scatter sound, 
but also affect the sound velocity. If the boundary of 
the wake is sufficiently sharp, some sound will be re¬ 
flected backward. Since the reflected sound rays will 
go predominantly in the backward direction, rather 
than out in all directions, the resulting wake echo 
can be quite high even though the coefficient of re¬ 
flection is not very great. For the bubble densities 
found in destroyer wakes, and summarized in Table 
2, the reflection coefficient found from equation (85) 
of Chapter 28 is less than 0.4 X 10 _fi and therefore 
quite negligible. It is possible that higher bubble 
densities might be present in the highly reflecting 
wakes of the vessels discussed in Section 33.4, but 
this seems unlikely. These high values (see Table 5 of 
Chapter 33) were found in early measurements in 
shallow harbor waters and have not been reproduced 
in later, more accurate determinations on wakes of 
the same vessels. For example, early measurements 











538 


ROLE OF BUBBLES IN ACOUSTIC WAKES 


on a 40-ft motor launch gave a value of 2 db for W; 
later measurements on the same ship, with more 
standard equipment, gave a value of —21 db. In view 
of the failure of the later measurements to reproduce 
the early high values, these early values can probably 
be neglected. Until more detailed information is avail¬ 
able it may therefore be assumed that on the average 
the wake strength of large moving surface vessels, 
measured with long pulses, are all close to the theo¬ 
retical maximum values found from equation (5); 
that is, about — 16 db for rms amplitudes and — 10 db 
for average peak amplitudes, at 24 kc. 

The high values of IT found at 60 kc are not easily 
explained. These values are believed to be less ac¬ 
curate than those at 24 kc, since the equipment had 
not yet been wholly standardized. It is perhaps sig¬ 
nificant that in one of the most careful tests — the 
measurements on the wake of the Scripps discussed 
in Section 33.4 — the value of IT at 60 kc was actu¬ 
ally less than that at 24 kc (see Table 9 of Chapter 
33). Moreover, use of this same underwater sound 
equipment in measurements of target strengths of 
submarines has yielded results at 60 kc which are also 
10 to 20 db above the 24 kc results, in contradiction 
to theoretical expectations (see Sections 21.4.3 and 
23.6.2). It is also important that measurements on 
submarine wakes, made with different equipment and 
discussed below, show a decrease of IT with increas¬ 
ing frequency rather than an increase. It is possible 
that the bubble density at 60 kc is sufficiently high 
and the wake boundary sufficiently sharp that specu¬ 
lar reflection of sound at the wake boundary is suf¬ 
ficient to account for the high wake echoes observed; 
this possibility has not been investigated theoreti¬ 
cally. Until the high wake strengths found at 60 kc can 
be either explained or shown to be the result of ob¬ 
servational error, they will remain a serious discrep¬ 
ancy in the study of wakes. 

34.3.2 Submarines 

Values of W for submarines both submerged and 
surfaced are presented in Table 4 of Chapter 33. For 
surfaced submarines no estimates are available at 
20 kc; but at 45 kc, the value of W found for two sub¬ 
marines is —13 db. When 6 db is subtracted to give 
the wake strength in terms of the average intensity, 
this value is in close agreement with the maximum 
wake strength of about —16 db found at 24 kc. While 
no experimental data are available on the value of the 
damping constant 5 at 45 kc, Figure 1 suggests that 


the value of 10 log (a s /87rix e ) at 45 kc does not differ 
by more than a few decibels from its value at 24 kc. 
Thus it may be inferred that the wake strength for a 
surfaced submarine is quite comparable with that for 
any large moving surface vessel. The decrease of IT 
shown at 60 kc in Table 4 of Chapter 33 is probably 
not significant. 

This same Table 4 shows that the wake of a sub¬ 
merged submarine is a much poorer reflector than the 
wake of a surfaced submarine. Since the wake 
strength is less than its maximum value, IT should 
vary with the bubble density, and therefore with 
submarine depth and speed. While the measurements 
are not very conclusive, they indicate that for a sub¬ 
marine at 6 knots and a depth of 45 to 90 feet, IT is 
about —25 db at 45 kc; this estimate may well be in 
error by as much as 5 db. As before, an additional 
6 db must be subtracted to convert to an intensity 
scale, giving —31 db for IT. If 10 log (3/85) at 45 
kc is taken from Table 5, equation (8) gives 

10 log u(R r ) = —31 — 0.8 — 10 log h 

— 10 log w — 19.6. (9) 

If the wake is 10 yd deep and 30 yd across, the bubble 
density u(R r ) is about 3 X 10 -8 , less than a hundredth 
of the values for destroyer wakes 1 minute old at 15 
knots found in Table 1; the assumed wake dimensions 
are somewhat uncertain, but any reasonable varia¬ 
tion of these figures would not change the order of 
magnitude of u(R r ). If the curve of u(R r ) against R r 
were the same as the typical curves for destroyers, 
the total fraction u of the wake volume occupied by 
air bubbles would be only about 1 X 10~ 9 . While no 
other quantitative measurements are available, prac¬ 
tical echo-ranging tests indicate that the bubble 
density decreases with increasing submarine depth 
as would be expected at the greater pressure. 

In the top 60 ft this decrease is about as rapid as 
would be expected from the experiments with model 
propellers. Both the transmission measurements dis¬ 
cussed above and the reflection measurements dis¬ 
cussed below show that with a 10-in. propeller all 
acoustic effects are much reduced at depths below 
30 ft. However, the reflection measurements indicate 
that the acoustic effects have largely disappeared at 
depths below 60 ft, while echoes from submarine 
wakes have been reported at greater depths. This 
difference may be due to lack of sensitivity of the 
acoustical equipment used for the model propeller 
experiments. Alternatively, the greater size of the 
full-scale propellers may enable the formation of 



DECAY OF WAKES 


539 


larger bubbles, which would persist longer at great 
depths. More complete measurements would be re¬ 
quired on submarine wakes of different depths before 
any detailed conclusions can be drawn as to the varia¬ 
tion of wake strength with depth. 

34.3.3 Model Propellers 

The studies of echoes from wakes of model pro¬ 
pellers, reported in Section 33.5, are not sufficiently 
detailed to compare with theoretical predictions. 
While echo levels were quantitatively determined, 
neither the geometry of the experiment nor the trans¬ 
ducer and hydrophone directivities are well enough 
known to make possible a prediction of the echo 
levels from the known properties of air bubbles. These 
measurements are of theoretical interest, however, 
because they provide information on the change of 
wake echoes with depth. This information has al¬ 
ready been discussed before. 

The data also provide an interesting qualitative 
confirmation of scattering theory. As is evident from 
Figures 9 and 10 of Chapter 33, the echo level re¬ 
mains relatively constant in the first 30 ft of increas¬ 
ing depth. In this same depth interval the attenua¬ 
tion in the wake, shown by Figures 3 and 4 of Chap¬ 
ter 32, decreases from a high value near the surface 
to less than 5 db at 30 ft. This behavior is in accord 
with equation (3); this equation predicts that as long 
as the transmission loss is more than a few decibels, 
the amount of sound scattered from a collection of air 
bubbles in water will be independent of the density 
of bubbles in the water. 

34.4 DECAY OF WAKES 

The observations on the decay rate of a wake’s 
acoustic properties should be consistent with the rate 
of disappearance of bubbles, if bubbles are actually 
responsible for scattering and attenuation of sound 
by wakes. Although optical measurements of the 
bubble density concentration in wakes have been 
contemplated, at present there are not available any 
nonacoustic observations of the rate of decay of 
bubbles. Neither does physical theory permit pre¬ 
dicting the rate of wake decay. As set forth in Section 
27.2, the turbulent internal motion may be the factor 
which determines the “life-time” of wakes. However, 
an adequate theoretical analysis of the effect of tur¬ 
bulence on the rate of rise of bubbles in wakes is 
still lacking. 


Pending the solution of this fundamental problem, 
the equations (49) and (50) of Chapter 33 suggest a 
partial test of the theory of decay of acoustic wakes. 
These equations established a quantitative relation 
between the decay rate of the wake strength and that 
of the total transmission loss across the wake; more¬ 
over, they do not involve any quantities which are 
unknown or difficult to determine. With this test in 
mind, simultaneous observations of the decay rates 
dH w /dt and dW/dt were made for a number of wakes 
laid by the E. W. Scripps, on November 28, 1944; 
these experiments have already been described, and 
the results were summarized in Table 2 of Chapter 32 
and Table 9 of Chapter 33. The results, though in¬ 
sufficient to verify the relationship predicted theo¬ 
retically, do not seem to be inconsistent with it. 

According to the discussion in Section 33.1.4, the 
following relation should hold for short pulses and 
fresh wakes: 

dW _ [O.46e- 0 46 "” ro/ ’ u ] r 0 dH w r 0 dH w 

dt Ll - e-° MH " r ° /w ] w dt ~ w dt ‘ { W) 

The factor F in brackets can be read from Figure 4 in 
Chapter 33, using //„,r 0 /w as argument; r 0 was equal 
to 2.4 yd, as 3-msec signals were used. The width of 
the Scripps wake is 45 yd at the age of 5 minutes; 
hence r 0 /w is about 0.05. The results of the numerical 
test of equation (10) are presented in Table 7. The 
observed and computed ratios ( dW/dt)/{dH w /dt ) 


Table 7. Observed and predicted decay rates. 



Frequency 

in kc 


24 

60 

clH„. „ 

—— in db per minute 

0.7 

0.4 

dW . ,, . . 

—— in db per minute 
clt 

1.5 

0.7 

dW !dH w , 

—— / —r~ observed 
dt / dt 

—2 

-2 

H u at age of 5 minutes in db 

3.0 

4.5 

H u ,ra/w at age of 5 minutes in db 

0.15 

0.22 

F 

7 

4 

Fro/w 

1.05 

0.88 

dW /dHu 

-*/ ~dT computed 

~1 

-1 


seem to agree as to order of magnitude. Little more 
can be expected, considering the high sensitivity of 
the test following from the rapid variation of the 
function F with H w . 








540 


ROLE OF BUBBLES IN ACOUSTIC WAKES 


At any rate, equation (10) and the corresponding 
one for long pulses, resulting from putting r 0 /w equal 
to 1, seems to account qualitatively for the shape of 
curves obtained by plotting wake strength against 
wake age, as illustrated by Figure 8 in Chapter 33. 
Generally, W remains constant during the first 5 
minutes after the wake has been laid, or it may even 
increase slightly. Thereafter W decreases linearly 
with time. However, the transmission loss H, v of the 
wake appears to decrease linearly with time right 
from the beginning of the wake. The explanation is 
that for young wakes the factor F in equation (10) is 
so much smaller than 1 that dW/dt equals 0. After 
about five minutes H w seems to have decreased to 
such an extent that F becomes of the order of one, or 
dW/dt and dH w /dt have reached the same order of 
magnitude. 

The observed rates of decay of wake echoes, noted 
in Chapter 33, are mostly between 1 and 2 db per 
minute; the much higher values recorded in Table 0 
of Chapter 33 may be caused by the rather shallow 
depth of the wake of the launch, as distinguished from 
the much deeper wakes of the larger surface vessels. 
In the interpolation formula for- H w — equation (10) 
in Chapter 32 — H w was assumed to decrease linearly 
with increasing time. An exponential decay would be 
more consistent with the observations of wake echoes, 
if equation (10) of this section is fulfilled; the meas¬ 
urements are not sufficiently accurate, however, to 
indicate which type of decay is actually followed. 

Thus it may be concluded that the observed decay 
rates for scattering and attenuation are mutually con¬ 
sistent, as far as the rather scanty evidence goes. 
Even if future wake observations would establish 
beyond doubt that equation (10) is satisfied, these 
results would by no means suffice to confirm the 


bubble hypothesis. It should be realized that equa¬ 
tion (10) represents a relationship of a quite formal 
nature and physically does not imply more than the 
plausible proposition that the acoustic effects of 
wakes are proportional to the volume density of some 
unspecified agent. 

The total time required for wakes to decay, how¬ 
ever, is consistent with the time required for small 
bubbles to disappear by resolution in sea water. The 
experiments discussed in Section 27.2.2 indicate that 
a bubble whose initial radius is 0.10 cm will disappear 
in about 30 minutes by gradual resolution of air back 
into the water. Turbulent motion is needed to keep 
air bubbles from reaching the surface but cannot pro¬ 
long the life of a wake beyond the time limit set by 
the resolution process. Thus 30 minutes is an upper 
limit for the life of an acoustic wake if the greatest 
air bubbles present are initially 0.10 cm in radius. 
Since bubbles of this size resonate to sound of 3 kc, 
the transmission loss observations described in Sec¬ 
tion 32.3.1 indicate that bubbles of this size are 
present initially. The observed length of time during 
which echoes are observed from a surface ship wake 
averages in the neighborhood of 30 minutes. Thus the 
observed rate of decay of acoustic wakes is at least 
generally consistent with the hypothesis that bubbles 
are responsible for the wake’s acoustic properties. In¬ 
formation on turbulence in wakes would be necessary 
for more detailed comparison. However, this general 
consistency lends added support to the “bubble 
hypothesis,” especially when added to the data al¬ 
ready discussed on (1) the variation with depth of the 
cross section for scattering and extinction, and (2) 
the value of the ratio <r s /<r e , and its variation with 
frequency, which affects the absolute values of the 
wake strength. 



Chapter 35 

SUMMARY 


T he wake of a moving ship scatters and attenu¬ 
ates sound. The following sections summarize 
existing data in the form of rules for predicting the 
geometry of acoustic wakes — their depths and 
widths, the attenuation of sound crossing wakes, and 
the scattering of sound from wakes. 

In some cases, these rules are based on few ob¬ 
servations. Moreover, the degree of reliability of most 
of the rules is difficult to assess, and an adequate ap¬ 
praisal of it in most cases can be reached only by 
study of the detailed expositions given in the pre¬ 
ceding chapters. 

35.1 WAKE GEOMETRY 

For surface ships, the depth h of an acoustic wake is 
approximately twice the draft of the wake-laying 
vessel, and is practically constant up to distances of 
at least 1,000 yd behind the ship (see Section 31.3.1). 
The depth of the wake laid by a surfaced submarine 
decreases from about 30 ft at a distance 100 yd be¬ 
hind the screws to about 20 ft at a distance astern of 
1,000 yd. The wake of a submerged submarine, run¬ 
ning at a periscope depth with a speed of 6 knots, 
reaches the ocean surface at distances astern greater 
than 100 yd, corresponding to a half-angle of diver¬ 
gence at the screws of about 5 degrees in the vertical 
direction (see Section 31.3.2). 

The width w of a wake increases with the range r 
behind the wake-laying vessel. For destroyer and 
destroyer escort wakes at distances astern greater 
than 100 yd, the wake fans out laterally in a regular 
manner, with the wake edges including a total angle 
of 1 degree (see Section 31.2). 

At distances less than 100 yd astern, the wake 
geometry is less regular and depends upon the speed 
of the destroyer in a complicated manner. This de¬ 
pendence may be represented by the following equa¬ 
tion: 

w* 

w = — r = 0.85 r, (1) 


which isvalid at distances astern r less than r*. For r* 
the values in Table 1, which were deduced from aerial 


Table 1 . Dependence of r* and w* on ship speed. 


Ship speed 
in knots 

r* 

in yards 

w* 

in yards 

16 

21 

18 

20 

39 

33 

25 

75 

64 

33 

93 

80 


photographs (see Section 31.1) of destroyer wakes, 
should be used. At distances astern greater than r*, 
one can compute the wake width by the equation 

w = w* + 0.017(r — /•*) , (2) 

using the same values of r* and w* as before. 

Acoustic determinations of the width of destroyer 
wakes (see Section 31.3.1) are much less accurate 
than the photographic measurements, and seem to be 
in moderate agreement with the predictions made on 
the basis of equation (2). 

The acoustic properties of the wake apparently 
vary with position inside the wake, although no defi¬ 
nite predictions can yet be made for a particular 
wake. Outside the boundaries established by the 
above relationships, the acoustic effects produced by 
the water are no greater than those typical of the 
.ocean with no wakes present. 

35.2 ABSORPTION BY WAKES 

When sound from a shallow projector is received on 
a shallow hydrophone, the transmission loss is in¬ 
creased by an amount H w if a wake is present between 
the projector and the hydrophone. This attenuation 
by the wake H„ may be expressed as 

II w — K e x (3) 

where K e is the attenuation coefficient in decibels per 
yard, and x is the length in yards of the sound path 
within the wake [see equation (2) of Chapter 32], 


541 






542 


SUMMARY 



Figure 1. Initial transmission loss across destroyer 
wakes. 


the wake, as bubbles with radii between R r and R, + 
dR, the attenuation coefficient K e in decibels per yard 
may be written [see equation (7) of Chapter 32] 

K e = 1.4 X 10 5 u(/? r ) . (4) 

In the wake less than 500 yd behind a destroyer or 
destroyer escort, the attenuation coefficient in the 
horizontal direction K e = H w /w is about 1 db per yd 
at 20 kc (see Section 32.3). If attenuation at other 
frequencies by the same wake is taken into account, 
the total amount of air is about 0.7 X 10~ 6 cu cm per 
cu cm of water in the wake of a destroyer at 15 knots, 
one minute after the passage of the vessel (see Tables 
1 and 2 of Chapter 28). 

For sound transmitted vertically upward and 
reflected back by the surface, thus traveling twice 
through the center of a destroyer wake about 20 ft 
thick, the attenuation coefficient is found from the 
equation 

„ H w 

Ke ~ 2 h ’ 

where H now denotes the two-way attenuation. The 



The attenuation coefficient K, is determined by the value of K e observed in this case is about 3 db per yd 
density of air in resonant bubbles of radius R r ■ If at 20 kc [see equation (13) of Chapter 32], 
u(R r )dR is the fraction of air present, in 1 cu cm of For sound transmitted along a horizontal path per- 



























ECHOES FROM WAKES 


543 



Figure 3. Distance astern in yards as function of wake age and speed of wake-laying vessel. 


pendicular to the wake axis, and within 10 feet of the 
surface, the transmission loss in destroyer wakes is 
given by the equation (see Section 32.3.1) 

H w = 1.5 {vf) 1 - 3.Of - a - 3.Of (5) 

where/is the frequency of the sound in kc, v the ship’s 
speed in knots, and f the age of the wake in minutes. 
When the projector is in the wake, the factor 1.5 in 
equation (5) should be replaced by 2.4; however, the 
value of H u in this case may be different for different 
projectors, since the sound output of the projector 
may be affected by the presence of the wake. Numeri¬ 
cal values of a and H u . resulting from equation (5) 
can be read from Figures 1 and 2, respectively; 
Figure 3 may be used to find the distances behind the 
wake-laying destroyer corresponding to different 
wake ages and ship speeds. 

For the wakes of large surface vessels at speeds be¬ 
tween 10 and 25 knots, the value of K e and H w are 
probably much the same as those given by equation 
(5) applying to destroyers and destroyer escorts. 

These values of K e and //„ are averages over the 
cross section of the wake and do not take into account 
possible large changes in these quantities with posi¬ 
tion in the wake. 

For transmission along wakes, equation (3) cannot 


be used for distances large compared to the depth of 
the wake, since scattered sound traveling other than 
straight paths through the wake may become im¬ 
portant. In particular, the transmission loss H w for 
propeller sounds observed directly behind a ship with 
a hydrophone at a depth of 10 to 20 ft is of the order 
of 10 to 100 db per kyd, for frequencies between 5 and 
60 kc. This low value may also be due in part to re¬ 
duction of the absorption coefficient K e at depths 
greater than 10 ft in the wake (see Section 32.4). 

35.3 ECHOES FROM WAKES 

The level E of the echo received from a wake can 
be determined from the so-called wake strength W 
using the equation 

E = S + W — 2 H + 10 log r F (6) 

where S is the level of the rms pressure on the axis of 
the projector, measured one yard from the projector 
in decibels above 1 dyne per sq cm; E is the rms pres¬ 
sure level of the echo, again in decibels above 1 dyne 
per sq cm; r is the range in yards from the projector 
to the wake, measured along the projector axis, il¬ 
lustrated in Figure 4; H is the transmission loss from 
the projector to the wake, defined as ten times the 































544 


SUMMARY 



Figure 4. Range from transducer to wake. 


logarithm of the ratio of rms pressures at a point one 
yard from the projector and at a point r yards away; 
and 'F is a wake index based on the transducer pattern 
which differs for different conditions. 

Equation (6) may be written 

E = S - 2H + T w (7) 

where T w is the target strength of the wake. Then 
T w is related to W, the target strength of a one-yard 
length of wake, by the equation 

T w = W + 10 log r + iF. (8) 

While in some ways it is convenient to picture the 
quantity W, called wake strength, as representing the 
target strength of a one-yard length of wake, Chapter 



Figure 5. Wake target strength as function of wake 
strength and range. 


33 — especially Sections 33.1.1 and 33.1.3 — should 
be studied for a full understanding of the physical 
meaning of wake strength. In particular, it should 
be noted that, according to equation (8), the target 
strength of the wake T w depends on the transducer 
pattern and on the range over which the echoes are 


received. Values for T w for different values of W and r 
may be found from Figure 5 if 'F is known. These re¬ 
lationships all assume that both the top and bottom 
of the wake are in the sound beam. 

Since the echo fluctuates, the rms pressure will not 
be constant within one echo. In this summary, the 
rms pressure is averaged within each echo and then 
over several echoes. If in each echo the peak rms 
pressure recorded is taken and then averaged over 
several echoes, the wake strength W and the echo 
level E will be about 6 db higher than the values 
given here. 

35.3.1 Long Pulses 

For pulses of duration r sufficiently long so that 
cr/2 exceeds the extension of the wake along the pro¬ 
jector axis, reasonably good predictions of the wake 
strengths of surface vessels and submarines can be 
made. 

If the attenuation H w across the wake exceeds a 
few decibels, the wake strength W is given by 

W = 10 log s + 10 log h (9) 

where h is the depth of the wake in yards, and -s is a 
function of frequency only, with the values indicated 
in Table 2. Thus the wake strength of an opaque wake 


Table 2. Dependence of reflection coefficient s on 
frequency. 


Frequency 
in kc 

10 log s 
in db 

1 

-21 

5 

-22 

8 

-23 

13 

-24 

19 

-25 

26 

-26 

36 

-27 

45 

-28 


10 yd deep is —16 db at 24 kc. As shown in Figure 1 
of Chapter 34 (see curve marked OBSERVATIONS), 
wake strengths at 60 kc are uncertain. Values at fre¬ 
quencies between 10 and 30 kc are probably correct 
to within about 3 db. A correction of 6 db must be 
added to the wake strengths computed from equa¬ 
tion (9) in order to make them apply to the peak 
amplitude of the average echo. For a moderately 
directional transducer, the value of 4> in the case of 
long pings is given by 

'F = J s + 8 , 


( 10 ) 

























ECHOES FROM WAKES 


545 


where J s is the surface reverberation index, defined 
by 

J . = 10 log [^- J"b(<f>)b'(<f>)d<f> ] > (11) 

where b(4>)b'(4>) is the composite pattern function of 
the echo-ranging transducer. For typical transducers, 
the surface reverberation index can be computed from 
the equation 

J s = 10 log y - 23.8 , (12) 

where 2y is the horizontal angular width, measured 
in degrees, of the sound beam between points down 
3 db from the axis. 

Thus in this simple case, the target strength of an 
opaque wake at 24 kc is given by 

T w = — 2G + 10 log h + 10 log r 

+ 10logy - 24+ 8 (13) 

This equation may be used to predict the initial 
strength of echoes received from the wakes behind 
surface vessels. 

If the total attenuation II w across the wake is less 
than 1 db, the wake strength W may be less than the 
value found from equation (10). The wake strengths 
of observed surface wakes are constant for about 2 to 
5 minutes, and thereafter decay at a rate of 1 to 2 db 
per minute; wake echoes can be observed, under good 
conditions, for 20 to 40 minutes after the passage of a 
vessel. These times are not inconsistent with what is 
known of the times required for air bubbles initially 
0.1 cm in radius to disappear by diffusion back into 
sea water. In this situation, T for a directional trans¬ 
ducer is 

'k = J s + 8 + 10 log sec j3 (14) 

where /3 is the angle between the projector axis and a 
line perpendicular to the wake axis. The increase of 
echo strength with increasing /3 predicted by equation 
(14) holds only so long as the ping length is greater 
than the extension AB of the wake along the pro¬ 
jector axis in Figure 4, and so long as the absorption 
loss along the path AB is less than 1 db. 

The wake strengths of submerged submarines at 
45 and 90 ft at speeds of G knots are about —30 db. 
Surfaced submarines appear to have about the same 
wake strength as that predicted for large surface 
vessels from equation (10) (see Section 33.1.2). 

35 . 3.2 Short Pulses 

When the pulse length cr/2 is less than the exten¬ 
sion AB of the wake along the projector axis in Figure 


4, the preceding equations are less useful. Although it 
is possible to predict wake strengths by adding to 
equation (10) a correction term depending on the 
signal length, the resulting values of W cannot simply 
be transformed into echo levels, using equation (7), 
or into target strengths, using equation (9), unless 
the echo ranging transducer is beamed perpendicu¬ 
larly at the wake. 

For short pulses, the wake strength W decreases 
with the decreasing ratio of the geometric pulse 
length r 0 measured in yards to the geometric width w 
of the wake, and can be predicted from the following 
equation 

W — 10 log s + 10 log h 

+ 10 log [1 - 10 _(/W5)/(ro/u,) ] + 0, (15) 

where h is the depth of the wake, measured in yards, 
and 10 log s — k is the same function of the frequency 
only as in equation (9), with the values indicated in 
Table 2. Numerical values of the third term on the 
right side of equation (15) can be read from Figure 6 



Figure 6. Wake strength term as function of attenu¬ 
ation and ratio of ping length to wake width. 


as a function of II w , the total attenuation across the 
wake, and of r 0 /iv, the ratio of signal length in yards 
to wake width. The wake strengths and echo levels 
computed from equation (15) refer to the peak of 
the average echo. 

Echo levels E and target strengths T w predicted 
for values of W computed on the basis of equations 
(15) and (11) should be quite satisfactory, provided 
that the sound is beamed at the wake nearly perpen¬ 
dicularly. For lack of anything better, the same pre¬ 
dictions may be used in case the sound beam strikes 
the wake obliquely. The expected discrepancies be¬ 
tween observations and predictions, for that case, are 
believed to be smaller than +5 db. 
























546 


SUMMARY 


35.3.3 Angular Variation of the 
Echo Level 

When a wake is insonified by a stationary trans¬ 
ducer and the echo is recorded by a different hydro¬ 
phone at several positions, the average echo level thus 
determined may show moderate variations with posi¬ 
tion of the hydrophone even after corrections for 
range to the wake, measured along the hydrophone 
axis, have been applied. This angular variation of the 
echo level has not been investigated experimentally; 
however, a simple theoretical estimate of the order of 
magnitude of this effect can be made for long pulses 
and may be useful [see equations (72), (73), and (76) 
of Chapter 28]. 

For pulses longer than the width of the wake meas¬ 


ured along the sound beam, the echo level should be 
proportional to 


cos a + cos 0 

if the wake is highly opaque (total attenuation across 
the wake more than a few decibels); and proportional 
to 

sec a (17) 

if the wake is acoustically transparent (total attenua¬ 
tion less than 1 db). In equations (16) and (17), 0 
denotes the angle between the transducer axis and a 
line perpendicular to the wake, as illustrated in 
Figure 4; a is the corresponding angle between the 
axis of the hydrophone and a line perpendicular to 
the wake. 




BIBLIOGRAPHY 


Numbers such as Div. 6-510-MI indicate that the document, listed has been microfilmed and that its title appears in the 
microfilm index printed in a separate volume. For access to the index volume and to the microfilm, consult the Army and 
Navy agency listed on the reverse side of the half-title page. 


1 . 

V/2. 

3. 


1 . 

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4. 


1 . 

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6 . 


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13. The Attenuation of Sound in the Sea, C. F. Eckart, NDRC 

6.1- sr30-1532, Report U-236, Project NS-140, UCDWR, 

July 6, 1944. Div. 6-510.22-M4 

14. The Influence of Thermal Conditions on Transmission of 
24-Kc Sound, Report U-307, Nobs-2074, Sonar Data 
Division, CUDWR, Mar. 16, 1945. Div. 6-510.4-M5 

15. Transmission of Sound in Sea Water. Absorption and Re¬ 
flection Coefficients and Temperature Gradients, E. B. 
Stephenson, Report S-1204, NRL, Oct. 16, 1935. 

Div. 6-510.22-MI 

16. Absorption Coefficients of Sound in Sea Water, E. B. 
Stephenson, Report S-1466, NRL, Aug. 12, 1938. 

Div. 6-510.222-MI 

17. Absorption Coefficients of Supersonic Sound in Open Sea 

Water, E. B. Stephenson, Report S-1549, NRL, Aug. 2, 
1939. Div. 6-510.222-M2 

18. Attenuation of Underwater Sound, F. A. Everest and H. T. 
O’Neil, NDRC C4-sr30-494, UCDWR, Revised July 30, 

1942. Div. 6-510.2-MI 

19. Attenuation of Sound in Sea Water, G. J. Thiessen, OSRD 
Liaison Office III-I-830, Report PS-162, CNRC, June 10, 

1943. Div. 6-510.22-M2 

20. “Ultrasonic Absorption in Water,” F. E. Fox and G. D. 
Rack, Journal of the Acoustical Society of America, 12, 
1941, p. 505. 

21. “Ultrasonic Interferometry for Liquid Media,” F. E. 
Fox, Physical Review, 52, 1937, p. 973. 

22. “Ultrasonic Absorption and Velocity Measurements in 
Numerous Liquids,” G. W. Willard, Journal of the Acous¬ 
tical Society of America, 12, 1940, p. 938. 

23. Acoustique-absorption des ondes ultra-sonares par I’eau, 
Note (1) De M. B. Biquard, Comptes Rendus, 1931, 


pp. 193, 226. ( The Diffusion and Absorption of Ultra 
Sonics in Liquids), B. Biquard and R. Lucas. 

24. “Absorpt ion of Supersonic Waves in Water and in Aqueous 
Suspensions,” G. K. Hartmann and H. Facke, Physical 
Review, 57, 1940, p. 221. 

25. “Absorptions Geschwindigkeits und Entgasungsmessun- 
genism Ultraschallge-beit,” C. Sorenson, Ann. d. Phys., 
26 [5], 1936, p. 121. 

26. “Absorptions of Ultra Sonic Waves in Liquids,” J. 
Claeys, J. Errera, H. Sack, Faraday Soc. Trans., 33, 1936, 
p. 136. 

27. The Extinction of Sound in Water, C. F. Eckart, NDRC 
C4-sr30-621, UCDWR, Aug. 31, 1941. 

Div. 6-510.11-MI 

28. Asdic Area Trials, G. E. R. Deacon and H. Wood, OSRD 

Liaison Office WA-669-14, British Internal Report 127, 
IIMA/SEE, Fairlie Laboratory, Great Britain, May 10, 
1943. Div. 6-570.21-M3 

29a. Biweekly Report covering period July 25-Aug. 7, 1943, 
NDRC 6.1-sr31-753, Project NO-140, WHOI, Aug. 11, 
1943, pp. 1-2. Div. 6-510.41-Ml 

29b. Biweekly Report covering period Sept. 5-18, 1943, 

NDRC 6.1-sr31-757, Project NO-140, WHOI, Sept. 22, 
1943, p. 2. Div. 6-510.41-M2 

29c. Biweekly Report covering period Sept. 19-Oct. 2, 1943, 
NDRC 6.1-sr31-758, Project NO-140, WHOI, Oct. 6, 
1943, pp. 1-2. Div. 6-510.41-M3 

29d. Biweekly Report covering period Oct. 3-16, 1943, 

NDRC 6.1-sr31-759, Project NO-140, WHOI, Oct, 20, 
1943, p. 3. Div. 6-510.41-M4 

29e. Biweekly Report covering period Oct. 17-30, 1943, 

NDRC 6.1-sr31-1060, Project NO-140, WHOI, Nov. 3, 
1943, p. 2. Div. 6-510.41-M5 

29f. Biweekly Report covering period Nov. 14-27, 1943, 

NDRC 6.1-sr31-1062, Project NO-140, WHOI, Dec. 1, 

1943, p. 2. Div. 6-510.41-M6 

30. ,4 Comparison of Calculated and Observed Intensities for 
Some Split Beam Sound Field Runs, R. R. Carhart and 

L. A. Thacker, Internal Report A-26, Oceanographic Sec¬ 
tion, UCDWR, Aug. 2, 1944. Div. 6-510.22-M5 

31. Sound Beam Patterns in Sea Water, NDRC 6.1-sr31-1730, 

WHOI, Oct, 10, 1944. Div. 6-510.11-M9 

32. Layer Effect, Echo-Ranging Section, R. W. Raitt and 

M. J. Sheehv, Internal Report A-35, UCDWR, Sept. 9, 

1944. Div. 6-510.41-M7 

33. Layer Effect at 24 Kc and 60 Kc, M. J. Sheehv, Internal 
Report A-51, UCDWR, Dec. 27, 1944. 

Div. 6-510.41-M8 

34. The Sound Field of Echo-Ranging Gear, NDRC 6.1- 
sr30-1206, Report U-113, UCDWR, Oct. 1, 1943. 

Div. 6-510.22-M3 

35. Sound-Ranging Experiments at Key West, July 23-30, 

1941, M. Ewing, OSRD 725, NDRC C4-sr31-130, 
WHOI, May 23, 1942. Div. 6-570.21-MI 

36. Asdic Area Trials, G. E. R. Deacon and H. Wood, OSRD 

Liaison Office WA-669-14, British Internal Report 127, 
HMA/SEE, Fairlie Laboratories, Great Britain, May 10, 
1943. Div. 6-570.21-M3 



BIBLIOGRAPHY 


549 


Chapter 6 


1. Attenuation of Sound in the Sea, C. F. Eckart, NDItC 

6.1- sr30-1532, Report U-236, Project NS-140, UCDWR, 

July 6, 1944. Div. 6-510.22-M4 

2. Some Evidence for Specular Bottom Reflections of 24 -Kc 

Sound, R. R. Carhart, Report A-17, San Diego Labora¬ 
tory, UCDWR, June 9, 1944. Div. 6-510.5-M2 

3. Bottom Sediment Charts [for the guidance of submarines J, 
The Hydrographic Office, July 1944. 

4. Bottom Reverberation. Dependence on Frequency, NDRC 

6.1- sr30-677, Report U-79, UCDWR, June 16, 1943. 

Div. 6-520.21-Ml 

5. Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- 
sr30-401, Report U-7, UCDWR, Nov. 23, 1942. 

Div. 6-520-M2 

6. Sonar and Submarine Diving: Monthly Progress Report 
for June 1945, Nobs-2083, WHOI, July 11, 1945, pp. 2-4. 

Div. 6-530.22-M21 

7. Transmission of 24-Kc and 60-Kc Sound in Very Shallow 
Water, M. J. Sheehy, Internal Reports A-31 and A-31a, 
UCDWR, Aug. 26 and Oct. 23, 1944. 

Div. 6-510.221-MI 
Div. 6-510.221-M2 

8. Acoustic Properties of Mud Bottoms, G. P. Woollard, 

WHOI, Dec. 6, 1944. Div. 6-510.5-M4 

9. Long Range Sound Transmission, M. Ewing and J. L. 
Worzel, Interim Report 1, Nobs 2083, WHOI, Aug. 25, 
1945. (See also Chapter 9 of this volume.) 

Div. 6-510.1-M4 


10. Some Sound Propagation Measurements in the Four¬ 
teenth Naval District, NDRC 6.1-sr30-1691, Report M-226, 
Project NS-140, Listening Section, UCDWR, June 19, 

1944. Div. 6-510.2-M6 

11. Some Shallow Water Sound Propagation Measurements 

in the Thirteenth Naval District, NDRC 6.1-sr30-1317, 
Report M-126, Projects NS-140, NS-163, Listening Sec¬ 
tion, UCDWR, Oct. 26, 1943. Div. 6-510.2-M2 

12. Transmission of Continuous Sound, Biweekly Report 

Covering Period January 23 to February 5, 1944, NDRC 

6.1-sr30-1233, Project NS-140, Report U-176, CUDWR, 
Feb. 11, 1944, pp. 9-12. Div. 6-510.2-M3 

13. Transmission Survey Block Island Sound, W. B. Snow, 
H. B. Hoff, and J. J. Markham, NDRC 6.1-srl 128-1027, 
Report D12/R616, CUDWR-NLL, Mar. 16, 1944. 

Div. 6-510.2-M5 

14. Sonic Listening Aboard Submarines, NDRC 6.1-srll31- 
1885, Sonar Analysis Section, CUDWR-SSG, February 

1945. Div. 6-623.1-M8 

15. Transmission of Underwater Sound over a Sloping Bottom, 

R. R. Carhart and K. O. Emery, Internal Report A-39, 
UCDWR, Oct. 1, 1944. Div. 6-510.5-M3 

16. Transmission of Continuous Sound, Biweekly Report 

Covering Period from January 23 to February 5, 1944 
NDRC 6.1-sr30-1233, Project NS-140, Report U-176, 
UCDWR, Feb. 11, 1944. Div. 6-510.2-M3 

17. Sonic Listening Aboard Submarines, NDRC 6.1-srll31- 

1885, CUDWR-SSG, February 1945. Div. 6-623.1-M8 


Chapter 7 


1. The Sound Field of Echo-Ranging Gear, OSRD 2011, 

NDRC 6.1-sr30-1206, Report U-113, UCDWR, Oct. 1, 
1943. Div. 6-510.22-M3 

2. Amplitude Fluctuations of Transmitted and Reflected 
Sound Signals in the Ocean, M. J. Sheehy, Internal Re¬ 
port A-29, UCDWR, Aug. 17, 1944. Div. 6-510.3-M3 

3. Correlation of Simultaneous Transmission in Deep Water 

at Different Frequencies, M. J. Sheehy, Internal Report 
A-44, UCDWR, Oct. 28, 1944. Div. 6-510.222-M3 

4. Variation of Signal Amplitude after Transmission in the 

the Sea, M. H. Hebb and N. M. Blachman, HUSL, Dec. 
19,1944. Div. 6-510.11-M10 

5. Detection of an Echo in the Presence of Reverberation, 


C. F. Eckart, OSRD 173, NDRC C4-sr30-175, UCDWR, 
May 12, 1942. Div. 6-560.32-MI 

6. Lloyd Mirror Effect in a Variable Velocity Medium, 
R. R. Carhart, Memorandum for File 01.92, Report 
M-140, UCDWR, Oct, 23, 1943. Div. 6-510.111-Ml 

7. Measurements of the Horizontal Thermal Structure of the 

Ocean, N. J. Holter, Report S-17, USNRSL, Aug. 18, 
1944. Div. 6-540.4-MI 

8. Fluctuation of Transmitted Sound in the Ocean, Technical 

Memorandum 6, NDRC 6.1-srll3l-1883, Sonar Analysis 
Section, CUDWR, Jan. 17, 1945. Div. 6-510.3-M4 

9. Theoretical Discussion of Reverberation, C. L. Pekeris, 

OSRD 684, NDRC C4-sr20-097, CUDWR, May 29, 
1942. Div. 6-520.1-M7 


Chapter 8 


1. Underwater Explosives and Explosions, August 15 to Sep¬ 
tember 15, 1942, Section Bl, NDRC. Div. 2-130-MI 

2. Relative Pressure Measurements in Shock Wave from Small 
Underwater Explosions, M. F. M. Osborne and A. H. 
Taylor, Report S-2305, NRL, June 10, 1944. 

Div. 6-551-M11 

3. Underwater Explosives and Explosions, April 15 to May 15, 
1944, Report UE-21, Division 8, NDRC. Div. 2-130-MI 

4. Transmission of Explosive Impidses in the Sea, T. F. 

Johnston and R, W. Raitt, NDRC C4-.sr30-403, Report 
U-8, UCDWR, Dec. 2, 1942. Div. 6-510.23-M6 


5. A Study of the Transmission of Explosive Impulses in Sea 

Water, T. F. Johnston, OEMsr-30, UCDWR, June 25, 
1942. Div. 6-510.23-M4 

6. Underwater Explosives and Explosions, February 15 to 
March 15, 1944, Division 8, NDRC, Report UE-19. 

Div. 2-130-MI 

7. Supersonic Flow and Shock Waves, AMP Report 38.2R, 
OEMsr-945, AMG-New York University, August 1944. 

Div. AMP-101.1-M9 

8. Hydrodynamics, H. Lamb, Cambridge University Press, 
Sixth Edition, 1932, pp. 481-489. 



550 


BIBLIOGRAPHY 


Chapter 9 


1. Relative Pressure Measurements in Shock Waves from Small 
Underwater Explosions, M. F. M. Osborne and A. H. 
Taylor, Report S-2305, NRL, June 10, 1944. 

Div. 6-551-M11 

2. Development of Single Sweep Equipment for Impulse Work, 

T. F. Johnston, OSRD 766, NDRC C4-sr30-189, 
UCDWR, Apr. 29, 1942. Div. 6-510.23-M3 

3. The Use of Electrical Cables with Piezoelectric Gauges, 

R. H. Cole, Report A-306, OSRD 4561, OEMsr-596, 
Projects OD-03, XO-144, Division 2, NDRC, WHOI, 
January 1944. Div. 2-111.11-M4 

4. Nature of the Pressure Impulse Produced by the Detonation 

of Explosives Under Water. An Investigation by the Piezo- 
Electric Cathode-Ray Oscillograph Method, Report CB- 
01670-12, OSRD Liaison Office W-201-1E, Admiralty Re¬ 
search Laboratory, Teddington, England, November 

1942. Div. 6-510.23-MI 

5. Propagation of Steep-Fronted Sonic Pulses Through the 

Sea, OSRD Liaison Office W-215-5, Internal Report 66, 
HMA/SEE, Fairlie Laboratory, England, Mar. 17, 1942. 

Div. 6-510.23-M2 

6. The Error in the Measurement of Pressure in an Explosion 

Pressure Wave Due to Finite Gauge Size and to Inadequate 
Frequency Response of the Recording Amplifier, Report 
ADM/219/ARB, OSRD Liaison Office WA-4243-2C, 
Road Research Laboratory, Great Britain, February 
1945. Div. 6-510.23-M13 

7. Underwater Explosives and Explosions, February 15 to 
March 15, 1944, Report UE-19, Division 8, NDRC. 

Div. 2-130-Ml 

8. A Study of the Transmission of Explosive Impulses in 

the Sea Water, T. F. Johnston, OEMsr-30, NDRC 
UCDWR, June 25, 1942. Div. 6-510.23-M4 

9. Transmission of Explosive Impulses in the Sea, T. F. 

Johnston and R. W. Raitt, NDRC C4-sr30-403, Report 
U-8, UCDWR, Dec. 2, 1942. Div. 6-510.23-M6 

10. Solution of Acoustic Boundary Problems, Parts I to III, 

L. I. Schiff, University of Pennsylvania, Sept. 4, Oct. 7, 
and Nov. 2, 1943. Div. 6-510.1-M3 

11. Explosive Sound Waves in the Sea. Observations with a 

2500-cycle Moving-Coil Oscillograph, T. F. Johnston and 
R. W. Raitt, Memorandum M-10, OEMsr-30, UC'DWR, 
Sept, 16, 1942. Div. 6-510.23-M5 


12. Depth Charge Range Meter Tests, II. B. Hoff, G. R. Perry, 

et al., Memorandum D50/R1222, Project NS-238, 
CUDWR-NLL, Nov. 24, 1944. Div. 6-642.31-MI 

13. The experimental points for Figure 5 are taken from 
reference 7, but the theoretical curves for Figures 5 and 6 
are taken from a recent unpublished calculation made by 
UCDWR. 

14. Theory of Diffraction of Sound in the Shadow Zone, C. L. 
Pekeris, NDRC 6.1-sr20-846, CUDWR, May 5, 1943. 

Div. 6-510.11-M6 

15. The Sound Field of Echo-Ranging Gear, OSRD 2011, 
NDRC 6.1-sr30-1206, Report U-113, UCDWR, Oct. 1, 

1943. Div. 6-510.2-M3 

16. Propagation of Sound in a Medium of Variable Velocity, 
C. L. Pekeris, NDRC C4-sr20-001, XLL, Sept. 29, 1941. 

Div. 6-510.11-M2 

17. Hydrophone Calibration by Explosion Waves, J. L. Carter 
and M. F. M. Osborne, Report S-2179, NRL, Apr. 19, 

1944. Div. 6-510.23-M11 

18. Factors Affecting Long Distance Sound Transmission in 

Sea Water, G. P. Woollard, NDRC 6.1-sr31-426, OSRD 
1505, WHOI, Mar. 30, 1943. Div. 6-510.1-MI 

19. Bibliography and Brief Review of Published Material on 
the Physical Principles of Submarine Detection, M. F. 
Manning, NRDC C4, September 1941. 

20. Long Range and Sound Transmission, Interim Report 1, 
Mar. 1, 1944-Jan. 20, 1945, M. Ewing and J. L. Worzel, 
Nobs-2083, WHOI, Aug. 25, 1945. Div. 6-510.1-M4 

21. Deep Water Sound Transmissions from Shallow Explosions, 
J. L. Worzel and M. Ewing, WHOI, (n.d.). 

Div. 6-510.23-M14 

22. Explosion Sounds in Shallow Water, M. Ewing and J. L. 
Worzel, Nllls-38137, NOL and WHOI, Oct, 11, 1944. 

Div. 6-510.23-M 12 

23. Theory of Propagation of Explosive Sound in Shallow 

Water, C. L. Pekeris, OSRD 6545, NDRC 6.1-sr 1131- 
1891, January 1945. Div. 6-510.12-M5 

24. The Propagation of Underwater Sound at Low Frequencies 

as a Function of the Acoustic Properties of the Bottom, 
J. M. Ide, R. F. Post, and W. J. Fry, Report S-2113, 
NRL, Aug. 15, 1943. Div. 6-510.5-MI 

25. Theory of Characteristic Functions in Problems of Anom¬ 
alous Propagation, W. H. Furry, Report 680, MIT-RL, 
Feb. 28, 1945. 


Chapter 11 


1. Reverberation in Echo-Ranging: Part I, General Prin¬ 
ciples, T. H. Osgood, OSRD 807, NDRC C4-sr20-149, 
CUDWR, July 28, 1942. Div. 6-520-MI 


2. Reverberation in Echo-Ranging: Part II, Reverberation 
Found in Practice, T. H. Osgood. OSRD 1422, NDRC 
6.1-sr20-84C, Project NS-140, CUDWR, Apr. 14, 1943. 

Div. 6-520-M3 


Chapter 12 


1. Measurements of the Horizontal Thermal Structures of the 
Ocean, N. J. Holter, Report S-17, USNRSL, Aug. 18,1944. 

Div. 6-540.4-MI 

2. Theory of Sound, Lord Rayleigh, The Macmillan Com¬ 
pany, New York, 2, 1926, p. 126. 


3. The Discrimination of Transducers Against Reverberation, 

OSRD 1761, NDRC 6.1-sr30-968, Report U-75, UCDWR, 
May 31, 1943. Div. 6-520.1-M8 

4. Bottom Reverberation. Dependence on Frequency, NDRC 
6.1-sr30-677, Report U-79, UCDWR, June 16, 1943. 

Div. 6-520.21-Ml 



BIBLIOGRAPHY 


551 


5. Bottom Reverberation at 24 Kc. E. IF. Scripps Data, R. R. 
Carhart, Internal Report A-7, UCDWR, May 18, 1944. 

Div. 6-520.21-M3 

6. Mathematics of Physics and Chemistry, H. Margenau and 
G. Murphy, D. Van Nostrand and Company, New York, 
1943, p. 246. 

7. Multiple Scattering, C. F. Eyring, R. J. Christiensen, and 


C. F. Eckart, Memorandum for File No. 01.40, UCDWR, 
Apr. 18, 1942. Div. 6-520.11-M2 

8. Relation beltveen Scattering and Absorption of Sound, 

Memorandum for File No. 01.40x01.72, Report SAS-8, 
CUDWR-SSG, Dec. 11, 194 L Div. 6-520.11-M5 

9. Theory of Sound, Lord Rayleigh, The Macmillan Com¬ 
pany, New York, 2, 1926, p. 145. 


Chapter 13 


1. Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- 
sr30-401, Report U-7, UCDWR, Nov. 23, 1942. 

Div. 6-520-M2 

2. Ibid., p. 23. 

3. A System for Recording Reverberation as it Occurs in the 

Ocean, NDRC 6.1-sr30-1202, Report Mill, UCDWR, 
Aug. 28, 1943. Div. 6-520.2-MI 

4. Operational Procedures and Equipment Used in Sonar 
Sound Field Studies, NDRC 6.1-sr30-2024, Report L r -295, 
Project NS-140, UCDWR, Feb. 15, 1945, p. 8. 

Div. 6-510.2-M8 

5. Limitation of Echo Ranges by Reverberation in Deep 
Water , Report M-361, Nobs-2074, Sept. 20, 1945. 

Div. 6-520.22-M2 

6. Summary of the Calibration of the Reverberation Equip¬ 
ment, November 24, 1943 to February 23, 1945, T. H. 
Schaefer, UCDWR, Apr. 18, 1945. Div. 6-520.2-M4 


7. Bottom Reverberation at 24 Kc. E. IF. Scripps Data, R. R. 

•Carhart, Internal Report A-7, UCDWR, May 18, 1944. 

Div. 6-520.21-M3 

8. Operational Procedures and Equipment Used in Sonar 
Sound Field Studies, NDRC 6.1-sr30-2024, Report U-295, 
Project NS-140, UCDWR, Feb. 15, 1945, p. 18. 

Div. 6-510.2-M8 

9. Apparatus for Recording Reverberation in the Sea, L. N. 
Liebermann, OEMsr-31, WHOI, Feb. 23, 1945. 

Div. 6-520.2-M3 

10. Volume Reverberation. Scattering and Attenuation versus 

Frequency, OSRD 1555, NDRC 6.1-sr30-670, Report 
U-50, UCDWR, Apr. 13, 1943. Div. 6-520.3-MI 

11. Characteristics of Some Transducers Used by UCDWR, 
Report U-23, UCDWR, May 6, 1943. 

12. A Practical Dictionary of Underwater Acoustical Devices 
NDRC 6.1-sr20-889, CUDWR-USRL, July 27, 1943. 


Chapter 14 


1. Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- 
sr30-401, Report U-7, UCDWR, Nov. 23, 1942. 

Div. 6-520-M2 

2. Volume Reverberation. Scattering and Attenuation versus 

Frequency, OSRD 1555, NDRC 6.1-sr30-670, Report 
U-50, UCDWR, Apr. 13, 1943. Div. 6-520.3-MI 

3. Theory of Sound, Lord Rayleigh, The Macmillan Com¬ 
pany, New York, 2, 1926. 

4. Limitation of Echo Ranges by Reverberation in Deep Water, 
Report M-361, Nobs-2074, Sept. 20, 1945. 

Div. 6-520.22-M2 

5. Workbook for Prediction of Maximum Echo Ranges, Bureau 
of Ships, Navy Department, NavShips 900,055 (labeled 
NavShips 900,050), December 1944. 

6. Survey of Underwater Sound. Ambient Noise, V. O. 
Knudsen, R. S. Alford, and J. W. Ending, OSRD 4333, 
NDRC 6.1-1848, Report No. 3, Sept. 26, 1944. 

Div. 6-580.33-M2 

7. The Influence of Thermal Conditions on Transmission of 

2'i-Kc Sound, Report L : -307, Nobs-2074, UCDWR, Mar. 
16, 1945. Div. 6-510.4-M5 

8. Theoretical Physics, G. Joos, G. E. Stechert and Com¬ 
pany, New York, 1932, p. 581. 

9. The Sea Surface and its Effect on the Reflection of Sound awl 

Light, C. F. Eckart, Report M-407, Nobs-2074 UCDWR, 
Mar. 20, 1946. Div. 6-520.11-M6 

10. Scattering of Sound by the Surface of the Sea, L. I. Schiff, 
Project NS-140. Memorandum for file M-217, UCDWR, 
May 15, 1944. Div. 6-520.11-M4 


11. Solution of Acoustic Boundary Problems: Part I, L. I. 
Schiff, University of Pennsylvania, Sept. 4, 1943. 

Div. 6-510.1-M3 

12. Solution of Acoustic Boundary Problems: Part II, L. I. 
Schiff, University of Pennsylvania, Oct. 7, 1943. 

Div. 6-510.1-M3 

13. Solution of Acoustic Boundary Problems: Part III, L. I. 
Schiff, University of Pennsylvania, Nov. 2, 1943. 

Div. 6-510.1-M3 

14. Echoes from Swells, G. E. Duvall, Report A-43, UCDWR, 

Oct. 27, 1944. Div. 6-540-MI 

15. Multiple Scattering, C. F. Eyring, R. J. Christiensen, and 

C. F. Eckart, Memorandum for File 01.40, UCDWR, Apr. 
18,1942. Div. 6-520.11-M2 

16. The Short Range Spatial Pattern Measurements on the 
JK-SK4928 Transducer at 24 Kc, N. Most, Internal Re¬ 
port No. A-52, UCDWR, Jan. 5, 1945. 

Div. 6-510.221-M3 

17. The Effect of the Ship’s Roll on Echo Ranging, J. S. 
McNown and C. F. Eckart, NDRC 6.1-sr30-1205, Re¬ 
port M-114, UCDWR, Oct. 8, 1943. Div. 6-510.3-M2 

18. The Discrimination of Transducers Against Reverberation, 

OSRD 1761, NDRC 6.1-sr30-968, Report U-75, UCDWR, 
May 31, 1943. Div. 6-520.1-M8 

19. Computed Maximum Echo and Detection Ranges for Sub¬ 
marine Echo-Ranging Gear, W. B. Snow and E. Gerjuoy, 
NDRC 6.1-srll31, 1128-1688, CUDWR, July 1944. 

Div. 6-570-M2 



552 


BIBLIOGRAPHY 


Chapter 15 


1. Range Limitation in Shallow Water as Controlled by Bot¬ 
tom Character, State of Sea, and Thermal Structure, F. P. 
Shepard, Report A-10, UGDWR, May 22, 1944. 

Div. 6-520.21-M4 

2. Bottom Reverberation at 24 Ke. E. W. Scripps Data, R. R. 
Carhart, Report A-7, UCDWR, May 18, 1944. 

Div. 6-520.21-M3 

3. Bottom Reverberation, R. J. Christiensen, Internal Report 

A-5, UCDWR, May 16, 1944. Div. 6-520.21-M2 

4. Calculation of Sound Ray Paths Using the Refraction Slide 
Ride, NavShips 943, BuShips-NDRC, May 1943. 

5. The Short Range Spatial Pattern Measurements on the 
JK-SK4926 Transducer at 24 Kc, N. Most, Internal 
Report A-52, UCDWR, Jan. 5, 1945. Div. 6-501.221-M3 


6. Bottom Reverberation. Dependence on Frequency, NDRC 
6.1-sr30-677, Report U-79, UCDWR, June 16, 1943. 

Div. 6-520.21-M 1 

7. Maximum Echo Ranges in Shallow Water, Technical 
Memorandum 5, CUDWR, Oct. 21, 1944. 

Div. 6-570.1-M5 

8. Computed Maximum Echo and Detection Ranges for Sub¬ 
marine Echo-Ranging Gear, W. B. Snow and E. Gerjuov, 
NDRC 6.1-srl 131, 1128-1688, CUDWR, July 1943. 

Div. 6-570-M2 

9. Bottom Reverberation in Very Shallow Water, NDRC 6.1- 

sr30-1845. Report SM-249, Projects NS-140, NS-297, 
Aug. 18, 1944. Div. 6-520.21-M5 


Chapter 16 


1. Theory of Sound, Lord Rayleigh, The Macmillan Com¬ 
pany, New York, 1, 1926. 

2. Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- 
sr30-401, Report U-7, UCDWR, Nov. 23, 1942. 

Div. 6-520-M2 

3. The Detection of an Echo in the Presence of Reverberation, 

C. F. Eckart, OSRD 173, NDRC C4-sr30-175, I'CDWR, 
May 12, 1942. Div. 6-560.32-MI 

4. Reverberation and Scattering, Series I, Sonar Data, Report 

MR-345-1, Nobs-2074, UCDWR, July 1945, pp. 4-6. 

Div. 6-520.22-M1 

5. Reverberation and Scattering, Series /, Sonar Data, Report 

MR-365-1, Nobs-2074, UCDWR, September 1945, 

pp. 4-6. Div. 6-510.22-M7 

6. Fluctuations in Reverberation Due to Scattering Centers in 

Water, Part II, L. I. Schiff, University of Pennsyl¬ 
vania, June 5, 1943. Div. 6-520.11-M3 

7. Probability and its Engineering Uses, Fry. 

8. Fluctuation of Transmitted Sound in the Ocean, Technical 

Memorandum 6, NDRC 6.1-srl 131-1883, CUDWR, Jan. 
17, 1945. Div. 6-510.3-M4 

9. The Effect of the Ship’s Roll on Echo Ranging, J. S. 
McNown and C. F. Eckart, Report M-114, NDRC 6.1- 
sr30-1205, UCDWR, Oct. 8, 1943. Div. 6-510.3-M2 

10. Theory of Random Processes, II. Uhlenbeck, Report 454, 

MIT-RL, Oct. 15, 1943. Div. 14-125-M7 

11. Coherence of CW Reverberation, Memorandum for File 
No. 01.40, Report SAS-11, CUDWR, Dec. 20, 1944. 

Div. 6-520.1-M9 

12. The Fluctuations in Signals Returned by Many Inde¬ 
pendently Moving Scatterers, A. J. F. Siegert, MIT-RL, 
Report No. 465, Nov. 12,1943. Div. 14-122.113-M7 

13. The Appearance of the A Scope When the Pulse Travels 
Through a Homogeneous Distribution of Scatterers, A. J. F. 
Siegert, Report 466, MIT-RL, Nov. 9, 1943. 

Div. 14-124.2-M2 

14. “Stochastic Problems in Physics and Astronomy,” S. 
Chandrasekhar, Rev. of Mod. Phys., 15, January 1943. 


15. “Mathematical Analysis of Random Noise,” S. O. Rice, 
Bell System Technical Journal, 23, July 1944, p. 289. 

15a. “Mathematical Analysis of Random Noise,” S. O. Rice, 
Bell System Technical Journal, January 1945. 

16. The Extrapolatory Interpolation and Smoothing of Station¬ 
ary Time Series, N. Wiener, NDRC Progress Report No. 
19 to the Services, MIT, Feb. 1, 1942. 

17. Frequency Characteristics of Echoes and Reverberation, 

W. M. Rayton and R. C. Fisher, OSRD 4159, Project 
NS-140, NDRC 6.1-sr30-1740, Report U-244, UCDWR, 
Aug. 9, 1944. Div. 6-520.3-M2 

18. The Theory of Reverberation and Echo, C. F. Eckart, 
NDRC C4-sr30-005, UCDWR, July 7, 1941. 

Div. 6-520.1-MI 

19. Theoretical Discussion of Reverberation, C. L. Pekeris, 
OSRD 684, NDRC C4-sr20-097, CUDWR, May 29, 1942. 

Div. 6-520.1-M7 

20. Frequency Spread of Reverberation as Measured with the 

Periodmeter, Memorandum for File No. 01.40, Report 
SAS-15, Sonar Analysis Section, CUDWR-SSG, Jan. 17, 
1945. Div. 6-520.3-M5 

21. Frequency Characteristics of Reverberation, Memorandum 

for File No. 01.40, Report SAS-16, Sonar Analysis Sec¬ 
tion, CUDWR-SSG, Nov. 23, 1944. Div. 6-520.3-M4 

22. The Dependence of the Operational Efficacy of Echo-Ranging 
Gear on its Physical Characteristics, II. Primakoff and 
M. J. Klein, NDRC 6.1-srl 130-2141, Project NS-182, 
CUDWR-USRL, March 15, 1945. Div. 6-551-M14 

23. Frequency Modulation in Echo Ranging, C. F. Eckart, 
NDRC C4-sr30-236, UCDWR, July 21,' 1942. 

Div. 6-635.1-M3 

24. Observations of Echo Signals Obtained Using Variable 

Frequency Transmission, E. M. Macmillan, NDRC C4- 
sr30-208, UCDWR, July 4, 1942. Div. 6-510.3-MI 

25. Coherence and Fluctuation of FM Reverberation, M. J. 
Sheehy, Report A-37, UCDWR, Sept. 19, 1944. 

Div. 6-520.3-M3 



BIBLIOGRAPHY 


553 


Chapter 20 


1. The Theory of Sound, Lord Rayleigh, London, 1896. 

2. “On the Absorption of Sound Waves in Suspensions and 
Emulsions,” P. S. Epstein, Theodor von Kdrmdn Anni¬ 
versary Volume, California Institute of Technology, 
May 11, 1941, p. 162. 

3. H. Stenzel, Ann. d. Physik, Series 5, 41, 1942, p. 245. 

4. H. Reissner, Helvetia Physica Acta, 11, 1935, p. 140. 

5. The Acoustic Properties of Domes: Part II, H. Primakoff, 
NDRC 6.1-srl 130-1366, USRL, Feb. 18, 1944. 

Div. 6-555-M17 


6. Reflection and Scattering of Sound, H. F. Willis, OSRD 

Liaison Office WA-92 lOf, NDRC C4-brts-501, British 
Internal Report 50, HMA/SEE, Fairlie Laboratory, 
Great Britain, Dec. 20, 1941. Div. 6-530.1-MI 

7. Reflections from Submarines, M. J. Klein and J. B. Kellar, 

NDRC 6.1-srl 130-1376, Project No. 222, LTSIIL, Apr. 15, 
1945. Div. 6-530.1-M3 

8. General Information and Sketch Book for the Engine Room 

Personnel of German Submarines, Type VII C, U.S. Navy, 
DTMB, May 1942. Div. 6-530.22-MI 


Chapter 21 


1. An Analysis of Reflections from Submarines, NDRC 
6.1-srl 131-1846, File 01.80, Technical Memorandum 4, 
Sonar Analysis Section, CUDWR-SSG, Sept. 9, 1944. 

Div. 6-530.22-M9 

2. Reverberation Studies at 24 Kc, OSRD 1098, NDRC C4- 

sr30-401, File 01.40, Report U-7, Reverberation Group, 
UCDWR, Nov. 23, 1942. Div. 6-520-M2 

3. Listening Techniques, Biweekly Report Covering Period 

October 4 to October 17, 1942, NDRC C4-sr30-396, 
UCDWR, Nov. 7, 1942, p. 5. Div. 6-530.22-M2 

4. Target Strength of a Submarine at 24 Kc, G. E. Duvall, 

File 01.80, Internal Report A-4. Echo-Ranging Section, 
UCDWR, May 10, 1944. Div. 6-530.22-M6 

5. Data at 45 Kc on Echoes from a Diving Submarine and its 
Wake, [W. M. Rayton], Report M-172a, Project NS-141, 
NDRC 6.1-sr30-1475, Sonar Section, UCDWR, Mar. 3, 

1944 . Div. 6-530.22-M4 

6. Internal Waves, Biweekly Report Covering Period Jan¬ 

uary 21 to February 3, 1945, NDRC 6.1-sr30-2025, 
Report U-297, UCDWR, Feb. 10, 1945, p. 6. 

Div. 6-501.4-M3 

7. Relative Echo Intensity versus Aspect, F. E. Gilbert, Jr., 

and J. Iv. Nunan, Report P29/R789, CUDWR-NLL, 
Mar. 10, 1944. Div. 6-530.22-M5 

8. Sonar and Submarine Diving: Monthly Progress Report 
for May 1945, Report 3, Nobs-2083, WHOI, May 10, 

1945, p. 3. Div. 6-530.22-M19 

9. Sonar and Submarine Diving: Monthly Progress Report for 

June 1945, Report 4, Nobs-2083, WHOI, July 11, 1945, 
pp. 2-4. Div. 6-530.22-M21 


10. Measurements made with 26-Kc DSS on USS Cythera 
(Memorandum), C. A. Ewaskio, HL'SL, Feb. 21, 1945. 

Div. 6-632.422-M3 

11. Submarine Runs with Directional and Nondirectional 
Transmitting Beams, 26-Kc DSS on USS Cythera (Memo¬ 
randum), C. M. Clay, HUSL, June 18, 1945. 

Div. 6-632.422-M13 

12. Sound Ranges Under the Sea — 1944, OSRD 4400, 

NDRC 6.1-srll31-1880, Sonar Analysis Section, 
CUDWR-SSG, November 1944. Div. 6-500-M2 

13. Small Object Detection, Sonar Data: Monthly Progress Re¬ 
port, Series I, Report MR-323-1, Project NS-140, Nobs- 
2074, UCDWR, May 1945, pp. 9-10. Div. 6-530.22-M18 

14. Reflection of Sound from Targets, Sonar Data: Monthly 
Progress Report, Series I, Report MR-334-1, Nobs-2074, 
UCDWR, June 1945, pp. 10-12. Div. 6-530.22-M20 

15. Reverberation and Scattering, Sonar Data: Monthly 

Progress Report, Series I, Report MR-345-I, Nobs-2074, 
UCDWR, July 1945, pp. 4-6. Div. 6-520.22-MI 

16. The Attenuation of Sound in the Sea, C. F. Eckart, 
NDRC 6.1-sr30-1532, Report U-236, File 01.70, Project 
NS-140, UCDWR, July 6, 1944. Div. 6-510.22-M4 

17. Echoes of Very Short Pings from Submarines, W. M. 

Rayton, Report M-301, Project NS-140, Nobs-2074, 
UCDWR, Mar. 1, 1945. Div. 6-530.22-M16 

18. Surface Reflected Submarine Echoes, Report M-306, File 

01.80, Project NS-140, Nobs-2074, Echo-Ranging Sec¬ 
tion, UCDWR, Mar. 15, 1945. Div. 6-530.22-M17 


Chapter 22 


1. Reflection of Light from a Submarine Model, R. B. Tibby, 

Memorandum for File 02.30, Report M-61, UCDWR, 
May 12, 1943. Div. 6-530.23-MI 

2. Reflections from Submarines at Close Ranges. Model Ex¬ 

periments Using Optical Method, Project NO-222 and 
MIT Research Project DIC-6187, MIT-USL, Apr. 8, 
1944 Div. 6-530.23-M2 

3. Studies of Optical Reflections from Submarine Models: 
Part II, OSRD 3706, NDRC 6.1-srl046-1053, Project 


NS-222 and MIT Research Project DIC-6187, MIT- 
USL, Apr. 12, 1944. Div. 6-530.23-M3 

4. Studies of Optical Reflections from Submarine Models: 

Part II, OSRD 3706, NDRC 6.1-srl046-166S, File 07.10, 
Navy Project NS-222 and MIT Research Project DIC- 
6187, MIT-USL, Aug. 15, 1944. Div. 6-530.23-M4 

5. Measurement of Reflections from Submarines Using Models 

and High-Frequency Sound, J. B. Kellar, OSRD 4439, 
NDRC 6.1-srl 130-1834, Navy Project NS-140, USRL, 
Sept, 27, 1944. Div. 6-530.23-M5 



554 


BIBLIOGRAPHY 


Chapter 23 


1. An Analysis of Reflections from Submarines, NDRC 6.1- 
srl 131-1846, File 01.80, Technical Memorandum 4, 
Sonar Analysis Section, CUDWR-SSG, Sept. 9, 1944. 

Div. 6-530.22-M9 

2. Target Strength of a Submarine at 24 Kc, G. E. Duvall, 

File 01.80, Internal Report A-4, Echo-Ranging Section, 
UCDWR, May 10, 1944. Div. 6-530.22-M6 

3. Sonar Sound Field, Biweekly Report Covering Period 
September 17 to September 30, 1944, NDRC 6.1-sr30- 
1862, Report U-262, UCDWR, Oct. 5, 1944, pp. 4-5. 

Div. 6-530.22-M10 

4. Pillenwerfer Design, OSRD Liaison Office WA-328-16, 
British Internal Report 100, HMA/SEE, Fairlie Labora¬ 
tory, Great Britain, Sept. 15, 1942. Div. 6-651-MI 

5. Studies of Optical Reflections from Submarine Models: 

Part II, NDRC 6.1-srl046-1668, Navy Project NS-222 
and MIT Project DIC-6187, File 07.10, M1T-USL, Aug. 
15, 1944. Div. 6-530.23-M4 

6. Reflections from Submarines, M. J. Klein and J. B. Kellar, 

NDRC 6.1-srl 130-1376, Navy Project NO-222, USRL, 
Apr. 15, 1944. Div. 6-530.1-M3 

7. Listening Techniques, Biweekly Report Covering Period 

October 4 to October 17, 1942, NDRC C4-sr30-396, 
UCDWR, Nov. 7, 1942. Div. 6-530.22-M2 

8. Reverberation Studies at 24 Kc, OSRD 1098, NDRC C4- 

sr30-401, File 01.40, Report U-7, Reverberation Group, 
UCDWR, Nov. 23, 1942. Div. 6-520-M2 

9. Data at 45 Kc on Echoes from a Diving Submarine and its 
Wake, [W. M. Rayton], NDRC 6.1-sr30-1475, Memo¬ 
randum for File 01.50, Service Project NS-141, Report 
M-172-A, Sonar Section, UCDWR, Mar. 3, 1944. 

Div. 6-530.22-M4 

10. Sonar and Submarine Diving. Monthly Progress Report for 

June 1945, Report 4, Nobs-2083, WHOI, July 11, 1945, 
pp. 2-4. Div. 6-530.22-M21 

11. Measurements Made with 26-Kc DSS on USS Cythera 
(Memorandum), C. A. Ewaskio, HUSL, Feb. 21, 1945. 

Div. 6-632.422-M3 

12. Measurement of Reflections from Submarines Using Models 

and High Frequency Sound, J. B. Kellar, OSRD 4439, 
NDRC 6.1-srl 130-1834, Navy Project NS-140, USRL, 
Sept. 27, 1944. Div. 6-530.23-M5 

13. Relative Echo Intensity versus Aspect, F. E. Gilbert, Jr., 

and J. K. Nunan, Report P29/R789, CUDWR-NLL, 
Mar. 10, 1944. Div. 6-530.22-M5 

14. Submarine Runs with Directional and Nondirectional 
Transmitting Beams, 26-Kc DSS on USS Cythera (Memo¬ 
randum) C. M. Clay, HUSL, June 18, 1945. 

Div. 6-632.422-M13 

15. Sonar Submerged Submarine Wakes, [P. H. Hammond], 
BuShips Problem U2-9CD, Serial S-RS-96, Report 
ND11/NP22/S68, USNRSL, Aug. 9, 1944. 

Div. 6-540.31-M3 


16. General Information and Sketch Book for the Engine Room 

Personnel of German Submarines, Type VII C, U.S. Navy, 
DTMB, May 1942. Div. 6-530.22-Ml 

17. Studies of Optical Reflections from Submarine Models, 

Part I, OSRD 3706, NDRC 6.1-srl046-1053, Navy 
Project NS-222 and MIT Project DIC-6187, File 07.10, 
MIT-USL, Apr. 12, 1944. Div. 6-530.23-M3 

18. Change of Average Peak Echo Intensity with Changing 

Ping Length, Lyman Spitzer, Jr., Memorandum for File 
01.80, Report SAS-30, Sonar Analysis Section, CUDWR- 
SSG, Mar. 22, 1945. Div. 6-530.1-M4 

19. Preparation of Charts of Average Echo-Ranging Condi¬ 
tions, Biweekly Report Covering Period July 23 to 
August 5, 1944, NDRC 6.1-sr30-1745, Report U-248, 
Project NO-140, UCDWR, Aug. 10, 1944, p. 5. 

Div. 6-530.22-M7 

20. Preparation of Charts of Average Echo-Ranging Condi¬ 
tions, Biweekly Report Covering Period August 6 to 
August 19, 1944, NDRC 6.1-sr30-1750, Report U-253, 
Project NO-140, UCDWR, Aug. 23, 1944, p. 4. 

Div. 6-530.22-M8 

21. Reflectivity of Targets, Biweekly Report Covering Period 

October 1 to October 14, 1944, NDRC 6.1-sr30-1865, 
Report L-264, Project NS-140, UCDWR, Oct. 31, 1944, 
pp. 3-5. Div. 6-530.22-M11 

22. The Attenuation of Sound in the Sea, C. F. Eckart, NDRC 

0.1-sr30-1532, Project NS-140, Report U-236, File 01.70, 
UCDWR, July 6, 1944. Div. 6-510.22-M4 

23. Reflectivity of Targets, Biweekly Report Covering Period 

October 29 to November 11, 1944, NDRC 6.1-sr30-1874, 
Report L T -274, Project NS-140, UCDWR, Nov. 16, 1944, 
pp. 5-6. Div. 6-530.22-M 13 

24. The Influence of Thermal Conditions on Transmission of 
24-Kc Sound, Sonar Data Division, Problem 2A, Report 
U-307, Nobs-2074, UCDWR, Mar. 16, 1945. 

Div. 6-510.4-M5 

25. Internal IFaees, Biweekly Report Covering Period Jan¬ 
uary 21 to February 3, 1945, NDRC 6.1-sr30-2025, 
Report U-297, UCDWR, Feb. 10, 1945, p. 6. 

Div. 6-501.4-M3 

26. Echoes of Very Short Pings from Submarines, W. M. 
Rayton, Problem 2C, Report M-301, File 01.80, Project 
NS-140, Nobs-2074, CUDWR, Mar. 1, 1945. 

Div. 6-530.22-M 16 

27. Reflectivity of Targets, Biweekly Report Covering Period 
January 7 to January 20, 1945, NDRC 6.1-sr30-2021, 
Report L T -292, Project NS-140, UCDWR, Jan. 26, 1945. 

Div. 6-530.22-M 14 

28. Origin of Nearest Echo, W. E. Benton, G. M. Johnson, 
W. A. Jones, and R. J. W. Morrison, British Internal 
Report 209, OSRD Liaison Office WA-4297-1 HMA/ 
SEE, Fairlie Laboratory, Great Britain, Feb. 15, 1945. 

Div. 6-530.22-M15 



BIBLIOGRAPHY 


555 


Chapter 

1. Surface Vessel Target Strengths, Memorandum for File 
01.80, SAG-38, Sonar Analysis Group, CUDWR-SSG, 

July 5, 1945. Div. 6-530.21-M3 

2. Oscillograms of 23-Kc Echoes from a Destroyer and its 
Wake, [C. F. Eckart], Memorandum for File 01.50, Re¬ 
port M-141, UCDWR, Jan. 3, 1944. Div. 6-530.21-MI 

Chapter 

1. Sonar Submerged Submarine Wakes, P. H. Hammond, 

BuShips, Problem U2-9CD, Serial S-RS-96, Report 
ND11/NP22/S68, USNRSL, Aug. 9, 1944, modified as of 
Nov. 1, 1944. Div. 6-540.31-M3 

2. Laboratory Studies of the Acoustic Properties of Wakes, 

J. Wyman, W. Lehmann, and D. Barnes, NDRC 6.1- 
sr31-1069, Project NS-141, WHOI, March 1944. 

Div. 6-540.3-M3 

3. The Rate of Rise and Diffusion of Air Bubbles in Water, 

C. L. Pekeris, OSRD 976, NDRC C4-sr20-326, CUDWR- 
PAG, Oct. 22, 1942. Div. 6-540.21-M2 

4. Propagation of Sound through a Liquid Containing Bubbles, 

Chapti 

1. “On the Absorption of Sound Waves in Suspensions and 
Emulsions,” Paul S. Epstein, Theodor von Karman Anni¬ 
versary Volume, CIT, May 11, 1941, pp. 162-168. 

2. The Stability of Air Bubbles in the Sea and the Effect of 

Bubbles and Particles on the Extinction of Sound and 
Light in Sea Water, P. S. Epstein, NDRC C4-sr30-027, 

UCDWR, Sept, 1, 1941. Div. 6-540.21-MI 

3. Propagation of Sound through a Liquid Containing Bub¬ 

bles: Part I, General Theory, L. L. Foldy, OSRD 3601, 
NDRC 6.1-srl 130-1378, Project NS-141, USRL, Apr. 25, 
1944. Div. 6-540.22-M2 

4. Leslie L. Foldy, The Physical Review, 67, 1945, p. 107. 

5. Acoustic Properties of Gas Bubbles in a Liquid, Lyman 
Spitzer, Jr., OSRD 1705, NDRC 6.1-sr20-918, CUDWR, 

July 15, 1943. Div. 6-540.22-Ml 

6. M. Minnaert, The London, Edinburgh and Dublin Philo¬ 
sophical Magazine and Journal of Science, 16, 1933, p. 

235. 


24 

!. Status Report on Task No. 5, Effect of Short Pulse Lengths 
and Receiver Bandwidth on Echo Ranging, R. W. Kirkland, 
Report 3510-RWK-HP, BTL, July 15, 1944. 

Div. 6-632.03-M5 

. Underwater Sound Reflecting Characteristics of Surface 
Ships, C. Shafer, Jr., Report 2320-CS-PD, BTL, Oct, 6, 
1944. Div. 6-530.21-M2 

27 

Part II, Experimental Results and Theoretical Interpreta¬ 
tion, E. L. Carstensen and L. L. Foldy, OSRD 3872, 
NDRC 6.1-srl 130-1629, Project NS-141, USRL, June 23, 
1944. Div. 6-540.3-M4 

. The Effect of Turbulent Motion on the Rate of Rise of 
Bubbles in a Wake, J. S. McNown, NDRC 6.1-sr30-731, 
File 01.50, Report U-25, UCDWR, Feb. 19, 1943. 

Div. 6-540.21-M3 

. Geometry on Surface Wakes and Experiments on Artificial 
Wakes, N. J. Holter, BuShips Problem U2-9CD, Report 
S-10, USNRSL, May 22, 1943. Div. 6-540.1-MI 

28 

7. E. Meyer and K. Tamm, Akustische Zeitschrift, 4, 1939, 
p. 145. 

8. Propagation of Sound through a Liquid Containing Bub¬ 

bles, Part II, Experimental Residts and Theoretical Inter¬ 
pretation, E. L. Carstensen and L. L. Foldy, OSRD 3872, 
NDRC 6.1-srl 130-1629, Project NS-141, USRL, June 
23, 1944. * Div. 6-540.3-M4 

9. Statistical Mechanics, R. H. Fowler, Cambridge Univer¬ 
sity Press, 1929, p. 154. 

10. The Internal Constitution of the Stars, A. S. Eddington, 
Cambridge University Press, 1929. 

11. Handbuch der Astrophysik, Julius Springer, Berlin, 1930. 

12. “On the Illumination of a Planet Covered with a Thick 
Atmosphere,” B. P. Gerasimovic, Bulletin de I’Observa- 
toire Central a Poidkovo (Russia), 15, No. 127, 1937, p. 4. 

13. A Textbook of Sound, A. B. Wood, The Macmillan Com¬ 
pany, 1941, p. 362. 


Chapter 29 


1. Thermal Wake Detection, D. H. Garber, R. J. Urick, and 
J. Cryden, Report S-20, USNRSL, Jan. 12, 1945. 

Div. 6-540.4-M2 

2. Reflection of Sound in the Ocean from Temperat ure Changes, 
R. R. Carhart, NDRC 6.1-sr30-960, Project NS-140, 
Report U-74, UCDWR, May 17, 1943. 

Div. 6-510.4-M3 

3. Theoretical Discussion of Reverberation, C. L. Pekeris, 

OSRD 684, NDRC C4-sr20-097, CUDWR-PAG, May 
29, 1942. Div. 6-520.1-M7 


4. The Geometry of Surface Wakes and Experiments on 

Artificial Wakes, N. J. Holter, Report S-10, USNRSL, 
May 22, 1943. Div. 6-540.1-MI 

5. Preliminary Measurements on the Acoustic Properties of 

Disturbed Water, E. Dietze, NDRC C4-sr20-205, USRL, 
Sept. 7, 1942. Div. 6-540.3-MI 

6. Propagation of Sound Through a Liquid Containing Bub¬ 

bles, Part II, Experimental Results and Theoretical Inter¬ 
pretation, E. L. Carstensen and L. L. Foldy, OSRD 3872, 
NDRC 6.1-srl 130-1629, Service Project NS-141, USRL, 
June 23, 1944. Div. 6-540.3-M4 




556 


BIBLIOGRAPHY 


Chapter 30 

1. Operational Procedure and Equipment Used in Sonar Project NS-140, Report U-295, UCDWR, Feb. 15, 1945. 

Sound Field Studies, NDRC 6.1-sr30-2024, Service Div. 6-510.2-M8 

Chapter 31 


1. Laboratory Studies of the Acoustic Properties of Wakes 

(Parts I and II), J. Wyman, W. Lehmann, and D. Barnes, 
NDRC 6.1-sr31-1069, Service Project NS-141, WHOI, 
March 1944. Div. 6-540.3-M3 

2. Thermal Wake Detection, D. H. Garber, R. J. Urick, and 
Joseph Cryden, Report S-20, LTSNRSL, Jan. 12, 1945. 

Div. 6-540.4-M2 

3. The Geometry of Surface Wakes and Experiments on 

Artificial Wakes, N. J. Holter, Report S-10, USNRSL, 
May 22, 1943. Div. 6-540.1-MI 

4. Sound Transmission Loss Through and Thickness of the 

Wakes of Antisubmarine Vessels, N. J. Holter, Report 
S-13, USNRSL, Nov. 22, 1943. Div. 6-540.32-M2 

5. Sound Transmission Through Destroyer Wakes, OEMsr-30, 
Project NS-141, Report M-189, UCDWR, Mar. 8, 1944. 

Div. 6-540.32-M3 


6. Chemical Recorder Traces of Submarine Wakes at Kc, 

Internal Report A-23, G. E. Duvall, UCDWR, July 18, 
1944. Div. 6-540.31-M2 

7. Reflectivity of Targets, Biweekly Report Covering Period 

October 15 to October 28, 1944, NDRC 6.1-sr30-1871, 
Project NS-140, Report U-271, UCDWR, Oct, 31, 1944, 
pp. 5-6. Div. 6-530.22-M12 

8. Wake of a Fleet-Type Submarine, W. M. Rayton and G. E. 

Duvall, Internal Report A-34, Echo-Ranging Section, 
UCDWR, Sept. 5, 1944. Div. 6-540.31-M4 

9. Sonar Submerged Submarine Wakes, P. H. Hammond, 
BuShips Problem U2-9CI), Code 940, Serial S-RS-96, 
Report ND11/NP22/S68, USNRSL, Aug. 9, 1944. 

Div. 6-540.31-M3 


Chapt 

1. Sound Transmission through Destroyer Wakes, OEMsr-30, 

Project NS-141, Report M-189, Listening Section, 
UCDWR, Mar. 8, 1944. Div. 6-540.32-M3 

2. Underwater Sound Output of Cruiser, Destroyer, and 

Aircraft Carrier, Report SM-268, UCDWR and MIT- 
USL, Oct, 28, 1944. Div. 6-580.2-M4 

3. Reflectivity of Targets, Biweekly Report Covering Period 

January 7 to January 20, 1945, NDRC 6.1-sr30-2021, 
Project NS-140, Report U-292, UCDWR, Jan. 26, 1945, 
pp. 5—6. Div. 6-530.22-M14 

4. The Geometry of Surface Wakes and Experiments on 

Chapter 

1. Acoustic Measurements on Surface Wakes in San Diego 

Harbor, R. R. Carhart and G. E. Duvall, OSRD 1628, 
NDRC 6.1-sr30-961, Report U-62, UCDWR, May 8, 
1943. Div. 6-540.32-MI 

2. The Discrimination of Transducers Against Reverberation, 

OSRD 1761, NDRC 6.1-sr30-968, Report U-75, UCDWR, 
May 31, 1943. Div. 6-520.1-M8 

3. Status Report on Task No. 5. Effect of Short Pulse Lengths 
and Receiver Bandwidth on Echo Ranging, Robert W. 
Kirkland, Report 3510-RWK-HP, BTL, July 15, 1944. 

Div. 6-632.03-M5 

Chapter 

1. Laboratory Studies of the Acoustic Properties of Wakes, 

J. Wyman, W. Lehmann, and David Barnes, NDRC 
6.1-sr31-1069, Project NS-141, WHOI, March 1944. 

Div. 6-540.3-M3 


Artificial Wakes, N. J. Holter, Report S-10, USNRSL, 
May 22, 1943. Div. 6-540.1-MI 

5. Sound Transmission Loss Through and Thickness of the 

Wakes of Antisubmarine Vessels, N. J. Holter, Report 
S-13, USNRSL, Nov. 22, 1943. Div. 6-540.32-M2 

6. Transmission of Sound Along IFafces, NDRC 6.1-srl046- 

1054, Project NS-141 and MIT Research Project DIC- 
6187, MIT-USL, July 26, 1944. Div. 6-540.32-M4 

7. Laboratory Studies of the Acoustic Properties of Wakes, 

(Parts I and II), J. Wyman, W. Lehmann, and D. 
Barnes, NDRC 6.1-sr31-1069, Project NS-141, WHOI, 
March 1944. Div. 6-540.3-M3 

33 

4. Preliminary Report on Echoes from a Diving Submarine 

and Its Wake, Project M-172, Report M-172, Sonar Sec¬ 
tion, UCDWR, Jan. 22, 1944. Div. 6-530.22-M3 

5. Data at fo Kc on Echoes from a Diving Submarine and its 
Wake, W. M. Rayton, NDRC 6.1-sr30-1475, Project 
NS-141, Report M-172a, UCDWR, Mar. 3, 1944. 

Div. 6-530.22-M4 

6. Laboratory Studies of the Acoustic Properties of Wakes, 

(Parts I and II), J. Wyman, W. Lehmann, and D. Barnes, 
NDRC 6.1-sr31-1069, Project NS-141, WHOI, March 
1944. Div. 6-540.3-M3 

34 

2. Reverberation Studies at 2J+ Kc, OSRD 1098, NDRC 6.1- 
sr30-401, Report U-7, UCDW T R, Nov. 23, 1942. 

Div. 6-520-M2 




CONTRACT NUMBERS, CONTRACTORS, AND SUBJECT OF CONTRACTS 


Contract No. 


Name and Address of Contractor 


Subject 


NDCrc-40 

Woods Hole Oceanographic Institution 
Woods Hole, Massachusetts 

Studies and experimental investigations in 
connection with the structure of the super¬ 
ficial layer of the ocean and its effect on the 
transmission of sonic and supersonic vibra¬ 
tions. 

Studies and investigations in connection w’ith 
the oceanographic factors influencing the 
transmission of sound in sea w'ater. 

OEMsr-20 

The Trustees of Columbia University in the 
City of New York 

New York, New York 

Studies and experimental investigations in 
connection w’ith and for the development 
of equipment and methods pertaining to 
submarine w’arfare. 

OEMsr-30 

The Regents of the University of California 
Berkeley, California 

Maintain and operate certain laboratories 
and conduct studies and experimental in¬ 
vestigations in connection with submarine 
and subsurface warfare. 

OEMsr-31 

Woods Hole Oceanographic Institution 
Woods Hole, Massachusetts 

Studies and experimental investigations in 
connection with the structure of the super¬ 
ficial layer of the ocean and its effects on 
the transmission of sonic and supersonic 
vibrations. 

OEMsr-287 

President and Fellow’s of Harvard College 
Cambridge, Massachusetts 

Studies and experimental investigations in 
connection with the development of equip¬ 
ment and devices relating to subsurface 
w'arfare. 

OEMsr-346 

Western Electric Company, Inc. 

120 Broadway 

New York, New York 

Studies and experimental investigations in 
connection with submarine and subsurface 
warfare. 

OEMsr-1046 

Massachusetts Institute of Technology 
Cambridge, Massachusetts 

Studies and experimental investigations in 
connection w'ith (1) underwater sound 
transmission and boundary impedance 
measurements; (2) ship sound surveys at 
high frequencies; (3) development of de¬ 
vices for the control of underwater sounds; 
and (4) development of intense underwater 
sound sources for special purposes. 

OEMsr-1128 

The Trustees of Columbia University in the 
City of New York 

New York, New York 

Conduct studies and experimental investiga¬ 
tions in connection with and for the de¬ 
velopment of equipment and methods in¬ 
volved in submarine and subsurface war¬ 
fare. 

OEMsr-1130 

The Trustees of Columbia University in the 
City of New York 

New’ York, New’ York 

Conduct studies and experimental investi¬ 
gations in connection with the testing and 
calibrating of acoustic devices. 

OEMsr-1131 

The Trustees of Columbia University in the 
City of New’ York 

New’ York, New' York 

Conduct studies and investigations in con¬ 
nection with the evaluation of the applica¬ 
bility of data, methods, devices, and 
systems pertaining to submarine and sub¬ 
surface warfare. 





SERVICE PROJECT NUMBERS 


The projects listed below were transmitted to the Executive Secretary, 
National Defense Research Committee [NDRC], from the Navy Depart¬ 
ment through the Office of Research and Inventions (formerly the Coor¬ 
dinator of Research and Development), Navy Department. These are 
the principal Navy projects relating to the physics of sound in the sea. 


Service Project 

Number 

Subject 

NO-163 

Cooperation with the Navy in harbor surveys 


and surveys of ambient underwater noise 


conditions in various areas. 

NO-222 

Acoustic reflection fields of submarines. 

NS-140 

Acoustic properties of the sea bottom. 

NS-140 

Range as a function of oceanographic factors. 

(Ext.) 


NS-141 

Acoustic properties of wakes. 






INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. 
For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


“Absorption cross section” of bubble, 
466 

Absorption effect in underwater sound 
transmission 

absorption coefficient, 97-100 
attenuation measurements, 102-105 
bubble formation, 465-467 
coefficient of attenuation, 100 
frequency ranges, 105-107 
thermal structure, 102 
transmission anomaly, 100-101 
wakes, 541-543 
Acoustic interference 
echoes, 377 
intensity, 168-170 
target strength measurements, 410 
Acoustic interferometer for sound ve¬ 
locity measurements, 17 
Acoustic measurements in underwater 
transmission, 243-244, 474-477 
Acoustic wakes 

see Bubbles in acoustic wakes; 
Wakes, acoustic 

Acoustical axis of sound projector, 26- 
27 

Adiabatic pressure changes during 
bubble formation, 461 
Aerial photographs in acoustic wake 
geometry, 494—495 
Air bubbles in acoustic wakes 
see Bubbles in acoustic wakes 
Airey phase of water waves, 232 
Anchored ships, target strength meas¬ 
urements, 424-425, 437 
Angular variation of echo level, 546 
Antinodes of stationary sound waves, 
33 

Aspect angle, target strength measure¬ 
ments, 388-393, 424 
Asymmetry effects on target strength 
measurements, 400-402 
Attenuation coefficient in sonic trans¬ 
mission 

bottom scattering, 320-321 
bubble formation, 469-470 
isothermal water, 100, 104-107 
shadow boundary, 124-125 
target strength measurements, 370, 
373, 411-413 

transmission anomaly, 129-131 
wake thickness, 503-504, 508-509 
Attenuation of sound 
bubble theory, 533-534 
explosions, 193-197 


frequency effects, 209-211 
long range transmission, 216-219 
propeller wakes, 510-511 
scattering layer, 299-301 
shadow zone, 67-68 
transmission anomaly, 100, 105-107 
wake theory, 503-504 
wave theory, 27-28 

Average layer effect in underwater 
sound transmission, 112 
Averaging methods for reverberation 
data, 278-280 


B-19 H magnetostrictive hydrophone, 
74 

Backward scattering coefficient of 
sound, 252, 266, 306, 335 
Backward scattering of sound, 254, 483 
Band method of averaging reverber¬ 
ation data, 279-280 
Bathythermograph 
classification, 92-95 
description, 76 
ray tracing, 60-63 
velocity-depth variations, 197-200 
Beam target strengths in echo ranging, 
415-417, 435-436 

Bell Telephone Laboratories (BTL), 
surface vessel target strengths, 
423-424 

Blade cavitation in acoustic wakes, 449 
“Blobs” in reverberation of sound, 335 
Bottom reverberation of sound, 264 
average intensities, 321-323 
data analysis, 319-321 
deep-water transmission, 86-87 
definition, 264 
description, 308-312 
frequency, 318-319 
grazing angle, 314-318 
refraction, 312-313 
scattering coefficients, 314, 319-321, 
338 

summary, 338-339 

Bottom scattering coefficients of sound, 
314, 319-321, 338 
Bottom-reflected sound 
attenuation coefficient, 103-104 
dispersion phenomena, 228-229 
normal modes theory, 222-224 
predictions of ray theory, 224 
ray intensity, 55-56 
reflection coefficient, 219-221 


shallow-water transmission, 137-138 
simple harmonic propagation, 224- 
227 

summary, 243 

supersonic frequencies, 140-141 
times of arrival, 221-222 
wave equation, 33-34 
Boundary conditions in sound propa¬ 
gation 

point source far from surface, 33-34 
point source near surface, 31-33 
reflection and refraction of plane 
waves, 30-31 

reflection from sea bottom, 33 -34 
target strengths, 353 
transition conditions, 28-31 
wake theory, 478 
wave equation, 13-14 
BTL (Bell Telephone Laboratories), 
surface vessel target strengths, 
423-424 

Bubbles in acoustic wakes 

absorption during bubble pulsation, 
464-467 

acoustic effects, 474-477 
attenuation, 469-470 
“bubble hypothesis”, 533 
buoyancy, 452-455 
damping constant, 467 
decay of wakes, 539-540 
echo intensities, 514-515 
entrained air, 455-457 
long pulses, 515-516 
multiple scattering, 470-473 
oscillograms, 186-190 
propeller cavitation, 449-452, 539 
reflection, 473-474 
scattering by an ideal bubble, 460- 
464 

scattering coefficient, 306-307 
short pulses, 516-519 
submarine wake strengths, 538-539 
surface vessel wake strengths, 537- 
538 

theory, 448, 467-469 
transmission loss, 503-504, 533-535 
wake echoes, 535-537 
Bulk modulus of a disturbed fluid, 12 
Buoyancy of bubbles in underwater 
sound, 452-455 

Burbling cavitation for bubble forma¬ 
tion, 449 

“Burning” process in underwater ex¬ 
plosions, 173-174 


559 


560 


INDEX 


Cable hydrophones, 74-75 
Calibration techniques for sound meas¬ 
urements, 492 

reverberation intensities, 277 
target strengths, 368-369 
transmission loss, 76-78 
Canadian National Research Council, 
attenuation measurements, 105 
Cathode-ray oscilloscope for acoustic 
wake measurements, 488-490 
Cavitation in bubble formation, 191, 
449-450 

CHARLIE bathythermograms, 93 
Chemical recorder traces in acoustic 
wake measurements, 484 
“Chirp” signal in echo-ranging gear, 23 
CN-8 crystal hydrophone, 74 
Coherence in sound reverberation, 335 
amplitude, 327-329 
intensity, 339 
transmission, 71 

Compression viscosity in attenuation of 
sound, 28 

Configurational averages for acoustic 
theory of bubbles, 468 
Conservation of energy law for second¬ 
ary sound pressure waves, 186- 
188 

Continuity law in sound wave propaga¬ 
tion, 8-10 

Continuous-flow bubble screens for 
acoustic measurements, 477 
Convex surface, target strength meas¬ 
urements, 359, 434 
“Cross-section” of bubble, 461 
CW pings, frequency analysis of rever¬ 
beration, 329-331 

Cylinder surfaces, target strength meas¬ 
urements, 360, 435 

Damped vibration, 28 
Damping constant, 467, 535-536 
Decay rate in sound transmission 
acoustic W'akes, 520-521, 539-540 
bottom reverberation, 322 
echo intensity, 526 
shock waves, 184-186 
surface reverberation, 337 
Deep-water reverberation of sound 
average reverberation levels, 304-306 
deep scattering layers, 282-284 
definition, 86-89 
echo ranging, 527-530 
frequency effects, 284-288 
multiple scattering effects, 303-304 
oceanographic conditions, 289 
ping length, 302 
range dependence, 289-302 
scattering coefficient, 306-307 
transducer directed downward, 281- 
28S 


transducer horizontal, 288-299 
volume reverberation, 281-282, 284- 
288, 335-337 

Deep-water transmission of sound 
see Transmission of sound, deep¬ 
water 

Density-pressure properties of a dis¬ 
turbed fluid, 11-12 
Depth effects in sound transmission 
bathythermograms, 92-95 
bottom reverberation level, 338 
corrections, 49-51 
ray diagrams, 89-90 
temperature gradients, vertical, 90- 
92 

thermocline transmission, 115-117 
volume reverberation, 282-284 
Destroyer wakes, air bubble hypothesis, 
534 

Detonation process in underwater ex¬ 
plosions, 173-175 
Diffraction of sound waves 
hypothesis, 201 
nonspecular reflection, 361 
pressure-time records, 204-206 
ray theory, 41 

shadow zones, 65-66, 200-201 
wave equation, 66-68 
Direction of sound propagation 
definition, 5-6 
directivity index, 72 
double source, 24-26 
pattern functions, 26-27 
point source, 24 

transducer patterns, 429-430, 522 
Doppler effect 

reverberation, 329-331 
wake measurements, 484 
Double layer effect in sound propaga¬ 
tion, 200 

Double sources of sound, 24-26 
Drift effect in echo variability, 376 

EBI-1 crystal transducer, 276 
Echo intensity measurements 
angular variation, 546 
definition of echo level, 434 
long pulses, 515-516 
short pulses, 516-519 
target strength, 347-348, 351, 377 
variability, 374-378 
Echo ranging 
equipment, 85 
frequency, 523 
projectors, 241 
pulse length, 522-523 
shallow-water, 321-323 
submarine wakes, 523-526 
surface vessel wakes, 526-530 
target strengths, 343-344, 376 
temperature gradients, 3—4 


thermocline, 109-110 
transducer directivity, 522 
wake measurements, 484, 490-493 
Echoes, wake 

beam echoes, 415-417, 435-436 
decay, 539-540 

off-beam echoes, 417-420, 436-437 
propellers, 539 
repeater, target training, 85 
source, 420-421 
submerged submarines, 437 
surface vessels, 437 
target strengths, 377, 435 
wake theory, 535-537, 543-546 
Eckart, self-correlation coefficient for 
sound intensity fluctuations, 166 
Eikonal wave equation in ray acoustics, 
44-45, 64-65 

Electromagnetic sources for sonic fre¬ 
quencies, 72 

Elongation phenomena of off-beam 
echoes, 418-420 

Entrained air in acoustic wakes, 455-457 
Equations for target strengths 
definition, 347 
derivations, 348-350 
reflected pressure, 355 
Equations of wave propagation, 8-14, 
43-45j 

boundary conditions, 13-14 
continuity, 8-10 

differential equations of rays, 45-46 
differential equations of wave fronts, 
43-45 

forces in a perfect fluid, 10-11 
initial conditions, 13-14 
motion, 10 
ray paths, 46-47 
state of fluid, 11-12 
wave equation, 12-13 
Equipment for reverberation measure¬ 
ments, 272-277 

Explosions, underwater, 173-235 
attenuation, 193-197 
bottom reflection, 219 
cavitation, 191 

deep sound channels, 213-216 
diffraction, 200-206 
Fourier analysis, 206-211 
long-range propagation, 211-213, 
216-219 

normal mode theory, 224-229 
predictions of ray theory, 222-224 
pressure waves, secondary, 186-190 
reflection coefficients, 220-222 
refraction, 197-200 
shallow-water experiments, 229-235 
shock fronts, 175-177, 182-184 
shock waves, 184-186 
summary, 173-175 
surface reflection, 190-191 



INDEX 


561 


transmission, 192-193 
variations, 211 
wave theory, 178-182 
“Extinction cross section” of bubbles, 
465-466 


Fathometer records for acoustic wakes 
submarines, 501-502 
surface vessels, 497-501 
thickness and structure, 486-488 
two-way vertical transmission loss, 
507-509 

Fermat’s theorem of reverberation in¬ 
tensity, 253, 269 

Fluctuations in sound transmission 
beam echoes, 436 
echo intensity, 377 
interference, 167-170 
magnitude, 158-160 
microstructure, lens action, 170-171 
off-beam echoes, 437 
probability distributions, 160-164 
reverberation, 324-327, 335, 339 
roll and pitch effects, 167-168 
sound pulses, 211 
space patterns, 167 
supersonic frequencies, 241 
time patterns, 164-167 
Fluid velocity of sound waves 
see Velocity of sound in water 
Fluorescein for acoustic measurements 
of wake-laying vessel, 491 
FM sonar, reverberation from wide¬ 
band pings, 75, 332-333 
Forced vibrations of bubbles, 461 
Forces in a perfect fluid, sound wave 
equation, 10-11 

Fort Lauderdale, Florida, target- 
strength measurements, 366, 368 
Forward reverberation of sound, 80 
Fourier theory in sound propagation, 
23, 36, 206-211, 329 
“Free vibrations” of bubbles, 461 
Frequency of sound 
attenuation, 138 
bottom reverberation, 338 
characteristics, 23-24 
deep-water reverberation, 284-288 
echo ranging, 408-410, 523 
narrow-band pings, 329-331 
penodmeter, 330 

shallow-water reverberation, 240, 
318-319 
sonic, 238-239 
supersonic, 238-239 
surface reverberation, 337 
target strengths, 433 
volume reverberation, 336 
wide-band pings, 332-333 


Fresnel zone theory of target strengths, 
356-360 

applications, 358 
convex surface, 359 
cylinder, 360 
method, 356-357 
sphere, 358-359 

Gaussian distribution of sound in¬ 
tensity fluctuations, 161-162, 
326 

Geometry of acoustic wakes 

see Wake geometry in sound trans¬ 
mission 

Grazing angle variation in reverber¬ 
ation of sound 

bottom scattering coefficients, 314- 
318 

transducer horizontal, 299-301 
Ground wave in sound transmission, 
230-232 

“Group velocity” of a wave train, 227 

Harbor detection equipment for sub¬ 
marine wakes, 443 

Harmonic waves in sound propagation, 
17-18, 22-23 

Heterodyned reverberation of sound, 
339 

“Hidden periodicities” of sound in¬ 
tensity fluctuations, 166-167 
“Highlight” in Fresnel zone theory of 
target strengths, 357, 358 
Horizontal transmission of sound 
beam echoes, 317-318 
bottom reverberation, 321-323, 339 
deep-water reverberation, 337-338 
transmission loss, 504-507 
transmission run, 79 
Hugoniot equation for shock fronts, 
180, 184 

Hull reflections of underwater sound, 
415 

Hull wake in sound transmission, 478 
Huyghen’s principle for reflected sound 
pressure, 356 

Hydrodynamic theory of bubble forma¬ 
tion, 449 

Hydrographic conditions for sound 
transmission anomalies, 119 
Hydrophone depth in sound transmis¬ 
sion, 72-74, 148-150 

Image effect in sound transmission, 95- 
97, 190 

Image interference in sound field in¬ 
tensity, 32-33, 163, 301 
Index of refraction, wave front equa¬ 
tions, 44 

“Instantaneous frequency” of rever¬ 
beration, 329-330, 339 


Intensity of echoes 
see Echo intensity measurements 
Intensity of sound, 6 
see also Fluctuations in sound trans¬ 
mission 

contours, 62-63 
experiments, 114-117 
formulas, 51-53 
interference effects, 168-170 
linear gradients, 57-58 
phase distribution, 37-38 
plane waves, 21 
rays, 65-66 

reverberation, 265-266, 334 
scattering, 532 
shadow zone, 65-68 
spherical waves, 21-22 
thermocline, 112-114 
transmission anomaly, 53-54, 58-59 
velocity-depth variation, 54-57 
wake measurements, 488-490, 504 
wave equation, 22 

Interference effects in sonic transmis¬ 
sion 

echoes, 377 
intensity, 168-170 
target strength measurements, 410 
Interferometer for sound velocity meas¬ 
urements, 17 

Inverse square law for underwater 
sound, 6-7, 237, 345-347 
Isothermal water, sound transmission 
absorption, 97-104 
attenuation coefficient, 104-107 
bottom reverberation, 313 
deep-water transmission, 238-239 
echo ranging, 109-110 
image effect, 95-97 
layer effect at 24 kc, 112-114 
layer effect at 60 kc, 117 
ray theory, 61 

short range transmission, 108-109 
temperature-depth pattern, 93 
thermocline depth, 114-117 
transmission loss, 107-108 
transmission runs, 110-111 
Iso velocity layer effect, ray acoustics, 
56-57 


Kennard’s theory of propagation of 
cavitation fronts, 191 

Khintchine’s theorem, self-correlation 
coefficient for sound intensity 
fluctuations, 167 

Lambert’s law for surface reverberation 
of sound, 300, 314 

Laminar cavitation in bubble forma¬ 
tion, 449 



5G2 


INDEX 


Law of conservation of energy for 
secondary sound pressure waves, 
186-188 

Law of motion for sound wave equa¬ 
tion, 10 

Law of similarity for shock waves, 182 
Layer effect at 24 kc, underwater sound 
transmission 
ray acoustics, 56-57 
theory, 112-114 

thermocline depth, 115-117, 238 
L T niversity of California studies, 114— 
115 

Layer effect at 60 kc, underwater sound 
transmission, 117 

Lens action of microstructure, sound in¬ 
tensity fluctuations, 170-171 
Listening equipment for wake measure¬ 
ments, 484 

Lloyd Mirror effect in wave acoustics, 
32-33, 299, 301 

Long Island area survey in sonic trans¬ 
mission, 154-156 

Long-range sound channel propagation, 
211-219 

deep channels, 213-216 
experimental results, 216-219 
introduction, 211-213 
Loops of stationary sound waves, 33 

Magnetostrictive effect in sound trans¬ 
mission, 5, 72 

Mean echo intensity for target strength 
measurements, 377-378 
Microdispersers for measuring damping 
constant in sound field, 467 
MIKE ba thy thermograms, 93-95 
Motion law for sound wave equation, 10 
Motion pictures of subsurface structure 
of wakes, 456 

Moving vessels, target strengths, 425- 
426, 437 

Multiple scattering of sound, 268-269, 
303-304 

NAN bathythermograms, 93-95 
Narrow-band pings, frequency analysis 
of reverberation, 329-331 
Naval warfare, acoustic wakes, 443-448 
Navy echo ranging 
see Echo ranging 

Newton’s second law of motion for 
sound wave equation, 10 
NK-1 type shallow depth recorder for 
acoustic wakes, 455 
Nodes of stationary sound waves, 33 
Noises, sinusoidal sound vibrations, 23 
Nonisothermal water, bottom rever¬ 
beration of sound, 321-323 
Nonresonant bubbles, acoustic meas¬ 
urements, 476, 477 


Nonspecular reflections of sound, 361- 
362, 410 

Normal mode theory of sound, 34-38, 
222-229 

bottom reflection, 222-224 
characteristic functions, 35-36 
dispersion phenomena, 225-228 
general waves, 36-37 
intensity of sound, 37-38 
plane waves, 34-36 
prediction of rays, 224-225 
pressure-time records, 228 


OAX transducers, 78 

Ocean bottoms, acoustics properties, 
139-141 

Oceanographic conditions for sound 
transmission 
bathythermographs, 76 
measurements, 243 
target strengths, 411-413 
wakes, 492-493 

Off-beam target strengths, 417-420, 
436-437 

One-way horizontal transmission loss, 
acoustic wakes, 504-506 

Optical experiments for target strength 
measurements, 379-381, 386, 
410 

Oscillograms for underwater sound data 
beam echoes, 415 
dispersion phenomena, 233-235 
echo intensities, 377 
explosive sound, 229-231 
ground wave phase of disturbance, 
230-233 

hydrophone output, 74-76 
pressure-time records, 204-206 
reverberation data, 278 
wake measurements, 488-490 

“Overtaking effect” in shock wave 
theory, 177, 183 


“Patch size” of acoustic wake, 479 
Pattern function for intensity of back¬ 
ward scattered sound, 254 
Peak echo intensity in target strength 
measurements, 373-374, 377- 
378 

Perfect fluid, law of forces, 10-11 
Periodmeter for frequency analysis of 
reverberation, 330-331, 339 
PETER bathythermograms, 93 
Phase constant in ray acoustics, 41 
Phase distribution in wave acoustics, 
37-38, 266 

Photographic Interpretation Center, 
Anacostia, wake acoustics, 494 


Physical parameters of acoustic wake 
strength, 514-519 
echo intensity, 514-515 
long pulses, 515-516 
short pulses, 516-519 
Piezoelectric effect in sound transmis¬ 
sion, 5, 72 

“piling-up” effect in long-range sound 
transmission, 218 
Pings in reverberation theory 
coherence, 327-329 
duration, 334 
narrow-band, 329-331 
short pulses, 326 
surface reverberation, 302 
volume reverberation, 336 
wide-band, 332-333 
Plane waves, sound propagation 
intensity of sound, 21 
normal mode theory, 34-36 
pressure versus fluid velocity, 19-20 
reflection and refraction, 30-31 
velocity of sound, 15-17 
wave equation, 14-15 
Point method of averaging reverber¬ 
ation data, 279-280 
Point source of sound 
boundary conditions, 33-34 
equation, 22, 31-32 
image interference effect, 32-33 
surface reflection, 32 
target strength, 353 
Poisson distribution of fluctuations of 
sound intensity, 326 
Power level recorders for underwater 
sound transmission measure¬ 
ments, 75 

Pressure of reflected sound wave, 352- 
355 

boundary conditions, 353 
mathematical analysis, 353-355 
physical analsyis, 355 
“Pressure pattern function” of sound 
receiver, 265 

Pressure versus fluid velocity of sound 
waves, 19-20 

Pressure waves (secondary) in sound 
propagation, 186-190 
oscillatory motion, 186-188 
spherical symmetry, 188-190 
Pressure waves (nonlinear) Riemann’s 
theory, 178-179 

Pressure-density properties of a dis¬ 
turbed fluid, 11-12 

Pressure-time curves of shock waves, 
184-186, 204-206, 228 
Probability coefficients for reverber¬ 
ation levels, 328 

Probability distributions of intensity 
fluctuations of sound field, 160- 
164 



INDEX 


563 


distribution functions, 160-162 
Gaussian, 161 
image interference, 163 
Rayleigh, 161-163 

Propagation of progressive waves, 14- 
15, 17-18 

Propeller wakes, sound transmission 
bubble density, 535 
bubble formation, 449 
scattering measurements, 530-532, 
539 

transmission loss, 510-511 
underwater explosions, 173-175 
Pulse length, sound measurements 
Fresnel zone theory, 362 
long pulses, 515-516 
short pulses, 516-519 
target strengths, 350-351, 404-408, 
432 

wakes, 522-523, 544-546 

QB crystal transducer, 275-276 
QCH-3 crystal transducers, 273-275, 
290-292 


Rankine-Hugoniot theory of shock 
fronts, 179-181 

Rarefractional shock waves in under¬ 
water explosions, 180-181 
Ray acoustics, 41-68 
curvature of ray, 46-47 
depth correction, 49-51 
diagrams, 59-60, 89-90 
eikonal wave fronts, 64-65 
general waves, 42-43 
intensity along a ray, 51-54 
long-range transmission, 216-219 
plotter for ray-tracing, 59 
ray patterns, equations, 45-46 
refraction, 197-200 
shadow zones, 65-68 
“sound channel” propagation, 211— 
216 

spherical waves, 41-42 
temperature-depth patterns, 60-63 
transmission anomalies, 58-59 
velocity-depth variation, 54-58 
vertical velocity gradients, 46-49 
wave front equations, 43-45 
Ray acoustics, theory of normal modes, 
222-229 

computations, 224-228 
dispersion phenomena, 225, 228-229 
predictability, 222-224 
Rayleigh’s sound scattering law 
deep-water reverberation, 288 
equation, 325-327, 481 
intensity fluctuations, 161-163, 169 
nonspecular reflection, 362 
radiation, long-wave, 464 


Receivers for underwater sounds, 73-76 
Reciprocity principle in sound propaga¬ 
tion, 38-39, 269-270 
Recommendations for sonar research, 
241-244, 339-340 
acoustic measurements, 244 
bottom reflection, 243 
oceanographic measurements, 243 
reverberation, 339-340 
surface reflection, 243 
velocity of sound, 242 
volume scattering, 242-243 
Reflected beam in linear gradient, ray 
acoustics, 55-56 
Reflected wave, 29 

Reflection and refraction of plane 
waves, 30-31 

Reflection coefficients for underwater 
sound 

long range transmission, 218-219 
ocean bottoms, 220-222 
sonic, 137-138 
supersonic, 140-141 
Reflection of sound 
bubble pulses, 473-474 
close ranges, 360 
submarines, 361, 386 
surface of water, 190-191, 373-374 
surface vessels, 437 
underwater targets, 352-355 
Refraction of sound 
bottom reflection, 138 
bottom reverberation, 312-313, 338 
bubbles, 473-474 
explosions, 197-200 
fluctuations, 170-171 
“Resolving time” for short-range sound 
propagation, 193 

Resonant frequency of an air bubble, 
462-464, 536-537 
Reverberation of sound 

see also Bottom reverberation of 
sound; Deep-water reverberation 
of sound; Surface reverberation 
of sound; Volume reverberation 
of sound 

analytical procedures, 278-280 
backward scattering coefficients, 266, 
335 

bottom levels, 310, 338-339 
coherence, 327-329, 335, 339 
deep-water levels, 335-338 
definition, 247, 334 
duration, 309 

equipment for measuring intensity, 
272-278 

Fermat’s principle, 269 
fluctuation, 158-160, 324-327, 335, 
339 

forward, 80 

frequency, 258-259, 329-333, 339 


intensity, 252-258, 265-266, 304-306, 
334 

level, 258-259, 334 
peak, 321-323 
ping length, 258-259 
properties, 247-249 
reciprocity theorem, 269-270 
recommendations for future research, 
338-339 

scattering, 250-252, 266-269 
strength, 259 
surface reflection, 270-271 
wakes, 492-493 

Riemann’s theory for sound waves of 
finite amplitude, 178-179 

Rigorous intensity in ray acoustics, 
65-66 

Roll and pitch effects on sound in¬ 
tensity fluctuations, 167-168, 
377 

Rough surface effects on reflection of 
sound, 361 


Salinity effect on sound velocity, 17 
Scattered sound 

see also Bubbles in acoustic wakes 
absorption, 242-243 
average levels, 304-306 
backward, 266, 335, 483 
bubble theory, 306-307, 470-473 
deep-w'ater reverberation, 286-288 
duration, 266-268 
multiple, 268-269, 303-304 
nonspecular reflection, 361 
propeller wakes, 530-532 
shadow zone, 125-129 
shallow-water reverberation, 316-317 
surface reverberation, 299-302 
temperature and velocity of w T akes, 
480-483 
theory, 250-252 

Screw wakes, sound transmission, 478 
Sea bottoms, acoustic properties, 139- 
141 

“Secondary sources” of sound, 356 
Self-con-elation coefficient for sound in¬ 
tensity fluctuations, 164-166, 
482 

Shadow boundary of sound 

attenuation coefficient, 124-125 
ray theory, 65-68, 89-90 
scattered sound, 125-129 
zones, 120-122, 200-206 
Shadow'ing effect in surface reverber¬ 
ation, 301 

Shallow-water reverberation 

see Bottom reverberation of sound 
Shallow-water transmission of sound 
see Transmission of sound, shallow- 
water 



564 


INDEX 


Ship draft and tonnage, effect on target 
strengths, 432 

Shock wave fronts, sound transmission, 
174-186 

law of similarity, 182 
Rankine-Hugoniot theory, 177, 179- 
181 

Riemann’s theory of waves of finite 
amplitude, 176-179 
structure and decay, 184-186 
thickness of pressure region, 182-184 
Short-range sound propagation, 108- 
109, 193-211 

diffraction hypothesis, 201-206 
Fourier analysis, 206-211 
pulse measurements, 193-197 
refraction effects, 197-200 
shadow zones, 200-201 
transmission variations, 108-109, 
211 

Similarity law for shock waves, 182 
Sinusoidal sound experiments, 192 
Slide rule for sound ray tracing, 59 
“Slipstreams” in sound transmission, 
478 

Snell’s law of refraction for bottom re¬ 
verberation of sound, 318 
Sonic transmission 

analysis of records, 83-84 
deep-water, 238-239 
frequency effects, 138-139 
listening gear, 87 
Long Island area survey, 155-156 
Pacific Ocean measurements, 156 
shallow-water, 240 
summary, 156-157 
“Sound channel” propagation 

deep sound channels, 213-216, 240 
experiments, 216-219 
long range transmission, 211-213 
surface sound channels, 239-240 
temperature gradients, 133-135 
Sound field measurements 

see Transmission loss measurements 
Sound propagation in liquid containing 
many bubbles, 467-477 
acoustical observations, 474-477 
reflection, 473-474 
scattering, 470-473 
theory, 467-469 
transmission, 469-470 
Sound range recorders for wake meas¬ 
urements, 484 

Sound transmission, underwater 

see Fluctuations in sound transmis¬ 
sion; Transmission of sound, 
deep-water; Transmission of 
sound, shallow-water 
Sources of sound 

see also Explosions, underwater 
directivity, 24-27 


echoes, 420-421 
frequency, 23-24 
levels, 347-348, 434 
transmission runs, 72-74 
Space pattern of fluctuation of sound 
intensities, 167 

Spectrum level in Fourier analysis of 
explosive sound, 208-209 
Specular reflection of sound 
beam echoes, 415-417 
convex surface, 434 
frequency factors, 410 
Fresnel zones, 356 
surface vessel, 430 
target strengths, 373-374 
Speed of ship, effects on target 
strengths, 402, 431 
Sphere target strengths 
definition, 434 
derivation, 348-350 
Fresnel zone theory, 358-359 
Spherical sound waves, 21-22, 41-42 
intensity, 21-22 

pressure versus fluid velocity, 20 
ray acoustics, 41-42 
wave equation, 18-19 
“Spines” of echoes in surface-reflected 
sound, 373-374 

Split-beam patterns in ray acoustics, 
61-62 

Standard reverberation level of sound, 
259 

Stationary waves in underwater sound 
see Normal mode theory of sound 
Still vessels, target strengths, 424-425, 
437 

Stoke’s hypothesis for attenuation of 
sound, 28 

Submarine reflectivity, 379-381, 386 
Submarine tactics in sound transmis¬ 
sion, 4 

Submarine target strengths, 388-421 
altitude angle, 393-397 
aspect angle, 388-393 
asymmetry, 400-402 
beam echoes, 413-417 
frequency, 408-410 
measurements, 397-400 
oceanography, 410-413 
off-beam echoes, 417-420 
orientation, 388 
pulse length, 404-408 
range, 402-404 
source, 420-421 
speed, 402 

Submarine wakes, acoustic measure¬ 
ments 

echoes, 523-526 
experiments, 501-502, 538-539 
Supersonic transmission 
data-analysis, 80-84 


deep-water, 238-239 
frequency effects, 138 
listening gear, 87 
sea bottoms, 139-141 
summary, 153 
transmission runs, 141-153 
velocity gradients, 142-150 
wind force, 152 

Surface reverberation of sound 
average levels, 304-306 
definition, 259 
elimination, 281 
grazing angle, 300-302 
index, 262 
intensity, 259-263 
“level” concept, 263-264 
multiple scattering, 303-304 
ping length, 302 
range, 289-299 
reflection effects, 270-271 
scattering coefficient, 306-307, 337 
summary, 336-338 
wind speed, 298-299 

Surface vessel target strengths 
aspect angle, 424-428 
deep-water transmission, 527-530 
frequency, 433 
introduction, 422 
measurements, 422-424 
pulse length, 432 
range, 426-431 
reflection, 437 
ship type, 432 
speed, 431 

wake echoes, 497-501 

Surface-reflected sound 
fluctuations, 377 
reverberation, 301 
short-range propagation, 196 
summary, 243 
transmission loss, 373-374 

Target strength measurement 
approximations, 352-353 
calibration errors, 368-369 
comparison of methods, 387 
computation, 358-360 
convex surface, 434 
cylinder, 435 
definition, 347-348 
echo variability, 374-378 
experiments, 363-366 
Fresnel zones, 356 -358 
introduction, 343 

Massachusetts Institute of Tech¬ 
nology, 379-381 
mathematical theory, 353-355 
Mountain Lakes, N. .1., 381 
nonspecular reflection, 361 
principles, 363-364 
pulse length, 350-351, 362 



INDEX 


565 


reflectivity, 353-355, 361, 386 
San Diego, 379 
scattering, 481-482 
spherical, 348-350, 434 
summary, 434-435 
surface vessels, 422-424 
transmission loss, 345-347, 369-374 
uses, 343-344 
wakes, 512-513 
wavelength effects, 386-387 
Targets, echo-ranging, 84 
Taylor Model Basin, sonic transmission 
experiments, 188, 456 
Temperature gradients in the ocean 
introduction, 3 

microstructure, 90-92, 482-483 
ray diagrams, 89-90 
refraction, 312-313 
60 kc transmission, 135 
surface effects, 239, 296-297 
velocity, 15-17 
wake structure, 441, 479—480 
Temperature gradients (negative) in 
the ocean 

attenuation coefficient, 124-125 
ray theory, 61-62 
shadow zones, 120-122, 125-129 
sharp gradients, 120-129 
60 kc transmission, 135 
sound channels, 131-133 
transmission anomalies, 118-120, 
122-123 

weak gradients, 129-133 
Temperature-depth patterns, ray dia¬ 
grams, 60-63 

Thermal microstructure for sound in¬ 
tensity fluctuations, 169-171 
Thermal wakes, sound transmission, 
441, 479-480, 496 
Thermocline, sound transmission 
see also Isothermal water, sound 
transmission 

below isothermal layer, 238 
ray theory, 61 

submarine target strengths, 411-413 
temperature gradients, 89 
Thermocouple recorder for sound ve¬ 
locity measurements, 17 
Thermodynamic law for absorption of 
sound, 464 

Thickness of acoustic wakes, 498-500 
Thickness of shock wave fronts, 177, 
182-184 

dissipation of energy, 183-184 
Hugoniot equation, 184 
Riemann overtaking effect, 183 
summary, 177 

“Time of arrival” of bottom-reflected 
sound pulses, 221-222 
“Time of rise” data for short-range 
sound propagation, 193 


Time patterns of sound intensity 
fluctuations, 164-167 
“hidden periodicities”, 166-167 
self-correlation coefficient, 164-166 
Training errors in echo-ranging on 
wakes, 491-492 

Transducers for acoustic measurements 
calibration, 78 
directivity, 522 
EBI-1; 276 
JK, 276 

QB-crystal, 275-276 
QCH-3; 273-275, 290-292 
reverberation intensities, 272-273,277 
target strengths, 429-430 
wakes, 492 

Transmission anomaly in underwater 
sound 

see also Attenuation of sound 
average, 122-123, 131-133 
bottom scattering, 319-321 
definition, 70-71, 237 
image effect, 95-97 
isothermal water, 100-104 
ray theory, 53-54, 58-59, 67-68 
supersonic, 147-150 
target strength, 369-371 
temperature gradients, 118-120 
Transmission loss measurements 
attenuation, 373, 503-504 
background, 3-4, 69-71 
bubble theory, 469-470 
echo runs, 84-85 
equipment, 76-78 
inadequacy, 372 
methods, 78-80 
observed echo ranges, 85 
oceanographic factors, 492-493 
one-way horizontal transmission, 504- 
506 

propagation along wakes, 509-510 
propeller wakes, 510-511 
receivers, 73-76 
sources, 72-74 
summary, 71-72 
supersonic frequencies, 80-84 
surface reflections, 373-374 
target strengths, 369-372, 411-413, 
430-431 

two-way horizontal transmission, 
506-507 

two-way vertical transmission, 507- 
509 

variation, 107-108 
wakes, 345-347, 504 
Transmission of sound, deep-water 
absorption, 97-104 
attenuation coefficient, 103-107 
bathythermograms, 92-95 
characteristics, 86-89 
echo-ranging trials, 109-110 


image effect, 95-97 
introduction, 86 
isothermal water, 95, 238-239 
layer effect, 112-117 
long-range experiments, 216-219 
negative temperature gradients, 118— 
120 

scattered sound in shadow zone, 125- 
129 

sharp temperature gradients, 120- 
125, 239 

short-range, 108-109 
60 kc transmission, 135 
sound channels, 133-135, 239-240 
thermocline, 110-111, 238 
transmission loss, 107-108 
variability of vertical temperature 
gradients, 90-92 

vertical temperature structure, 89-90 
weak temperature gradients, 129— 
133, 239 

Transmission of sound, shallow-water 
dispersion phenomena, 228-229 
experiments, 229-235 
reflection coefficient, 140-141 
sea bottoms, 137-140 
sonic, 154-157 
summary, 240-241 
supersonic, 139-140 
24 kc transmission, 141-143 
velocity gradients, 142-150 
wind force, 152-154 
Transmitted w T ave, 30 
Triangulation in long-range sound 
transmission, 219 
Triplane in echo ranging, 84 
Turbulence parameter for acoustic 
wakes, 452-455 

Two-way horizontal transmission loss 
in acoustic wakes, 506-507 

Underwater sound transmission, 236- 
244 

see also Fluctuations in sound trans¬ 
mission; Transmission of sound, 
deep-w r ater; Transmission of 
sound, shallow-w r ater 
recommendations for future research, 
241-244 

summary of definitions, 236-238 

Variability of echo intensity, 374-378 
“Variance of amplitudes” for sound in¬ 
tensity measurements, 237 
Variations in sound transmission 
short-range propagation, 211 
summary, 241 

transmission loss, 71, 107-108 
Velocity of sound in water 
bubble theory, 473-474 
microstructure, 482 



566 


INDEX 


pressure effects, 19-20 
ray equations, 46-49 
refraction effects, 197-200 
shallow-water transmission, 138 
summary, 242 

supersonic transmission, 142-150 
target depth correction, 49-51 
wake theory, 478^179, 480-483 
wave equations, 15-17 
Velocity-depth variation in ray acous¬ 
tics, 54-59 

beams in linear gradients, 54-58 
layer effect, 56-57 
transmission anomalies, 58-59 
Vertical temperature gradients in the 
ocean 

see Temperature gradients in the 
ocean 

Vertical transmission of underwater 
sound, 79, 507-509 

Viscosity (fluid) effects on sound in¬ 
tensity, 27-28 

Volume reverberation of sound 
average intensity, 255-256 
definition, 253-254 
depth, 282-284 
frequency, 284-288 
index, 259 
intensity, 256-258 
level, 258, 335-337 
range, 281 

scattering coefficient, 243, 286, 336- 
337 


Wake geometry in sound transmission 
aerial photographs, 494-495 
submarines, 501-502 


summary, 541-542 
surface vessels, 497-501 
target strength, 513-514 
widening measurements, 495-497 
Wake-laying vessel, acoustic measure¬ 
ments, 491 
Wakes, acoustic 
absorption, 541-543 
decay rate, 520-521, 539-540 
definition, 441 
echo ranging, 484, 543-546 
evaluation of research, 443-448 
fathometer records, 486-488, 497-501 
frequency, 523 

geometry, 494-495, 513-514, 541 
index, 513, 519-520, 543 
listening gear, 484 
long pulses, 515-516 
measurements, 490-492 
oceanographic effects, 492-493 
oscillograms, 488-490 
physical properties, 514-515 
propellers, 530-532 
pulse length, 522-523 
scattered sound, 480-483 
short pulses, 516-519 
sound range recorder, 484 
submarine, 501-502, 523-526, 538- 
539 

surface vessel, 526-530, 537-538 
target strength, 512-513 
temperature structure, 479-480 
theory, 541-546 
thickness, 498-501 
training errors, 491-492 
transducer directivity, 522 
velocity structure, 478-479 
widening rate, 495-497 


Water wave, sound transmission, 233- 


235 

Wave acoustics 

boundary conditions, 13-14, 28-34 
equation, 10-13, 43-45, 242 
equation of continuity, 8-10 
equations of motions, 10 
fluid viscosity effects, 27-28 
general waves, 36-37 
harmonic waves, 17-18, 22-23 
intensity, 20-22, 37-38 
mathematics, 39-40 
normal mode theory, 34-38 
plane waves, 14-17, 34-36 
pressure versus velocity, 19-20 
reciprocity principle, 38-39 
sources, 23-27 
spherical waves, 18-19 
Wave equation for shadow boundary, 
ray acoustics, 66-67 
Wave fronts, ray acoustics 
eikonal equation versus general equa¬ 
tion, 64-65 

equations, differential, 43-46 
general waves, 42-43 
spherical waves, 41-42 
Wave length effects on target strength 
measurements, 386-387 
Wavelets for reflected sound pressure, 
356 


Wide-band pings, frequency analysis of 
reverberation, 332-333 
Widening rate of acoustic wakes, 495- 


497 

Wind effects in sound transmission 
force, 337 
speed, 295-299 
supersonic, 150 




































GENERAL RESEARCH OFFICE 
THE JOHNS HOPKINS UNIVERSITY 
FT. L. J. McilAIR 
VMSHIIIGION, 2S, D.C. 


























































































































































































































































































































































